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321 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 322 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 64 "Integration of rational functions using partial fractions .. IV " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pet er Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Versio n: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 321 21 "Integrals of the form" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^n),x)" "6#-%$IntG6$*& \"\"\"F'),&*$%\"xG\"\"#F'*$%\"aGF,F'%\"nG!\"\"F+" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^2),x) = x/(2*a^2*(x ^2+a^2))+1/(2*a^3)" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF- F(F-!\"\"F,,&*&F,F(*(F-F(*$F/F-F(,&*$F,F-F(*$F/F-F(F(F0F(*&F(F(*&F-F(* $F/\"\"$F(F0F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c" "6#,&-% 'arctanG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "First note that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x/(x^2+a^2) ],x) = (1*`.`*(x^2+a^2)-x*`.`*2*x)/((x^2+a^2)^2);" "6#/-%%DiffG6$7#*&% \"xG\"\"\",&*$F)\"\"#F**$%\"aGF-F*!\"\"F)*&,&*(F*F*%\".GF*,&*$F)F-F**$ F/F-F*F*F***F)F*F4F*F-F*F)F*F0F**$,&*$F)F-F**$F/F-F*F-F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (a^2-x^2)/((x^2+a^2)^2);" "6#/%!G*&,&*$%\"aG \"\"#\"\"\"*$%\"xGF)!\"\"F**$,&*$F,F)F**$F(F)F*F)F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int((a^2-x^2)/((x^2+a^2)^2),x)=x/(a^2+x^2)+c" "6#/-% $IntG6$*&,&*$%\"aG\"\"#\"\"\"*$%\"xGF+!\"\"F,*$,&*$F.F+F,*$F*F+F,F+F/F .,&*&F.F,,&*$F*F+F,*$F.F+F,F/F,%\"cGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a^2-x^2)/((x^2+a^2)^2)+1/(x^2+a^2 ) = (``(a^2-x^2)+``(x^2+a^2))/((x^2+a^2)^2);" "6#/,&*&,&*$%\"aG\"\"#\" \"\"*$%\"xGF)!\"\"F**$,&*$F,F)F**$F(F)F*F)F-F**&F*F*,&*$F,F)F**$F(F)F* F-F**&,&-%!G6#,&*$F(F)F**$F,F)F-F*-F96#,&*$F,F)F**$F(F)F*F*F**$,&*$F,F )F**$F(F)F*F)F-" }{XPPEDIT 18 0 "``=2*a^2/(x^2+a^2)^2" "6#/%!G*(\"\"# \"\"\"*$%\"aGF&F'*$,&*$%\"xGF&F'*$F)F&F'F&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*a^2/((x^2+a^2)^ 2),x)=Int((a^2-x^2)/((x^2+a^2)^2),x)+Int(1/(x^2+a^2),x)" "6#/-%$IntG6$ *(\"\"#\"\"\"*$%\"aGF(F)*$,&*$%\"xGF(F)*$F+F(F)F(!\"\"F/,&-F%6$*&,&*$F +F(F)*$F/F(F1F)*$,&*$F/F(F)*$F+F(F)F(F1F/F)-F%6$*&F)F),&*$F/F(F)*$F+F( F)F1F/F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/(x^2+a^2)+1/a;" "6#/%!G, &*&%\"xG\"\"\",&*$F'\"\"#F(*$%\"aGF+F(!\"\"F(*&F(F(F-F.F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a) + c" "6#,&-%'arctanG6#*&%\"xG\"\"\"% \"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (1/((x^2+a^2)^2),x) = x/(2*a^2*(x^2+a^2))+1/(2*a^3);" "6#/-%$IntG6$*& \"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF-F(F-!\"\"F,,&*&F,F(*(F-F(*$F/F-F(,& *$F,F-F(*$F/F-F(F(F0F(*&F(F(*&F-F(*$F/\"\"$F(F0F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(x/a) +c" "6#,&-%'arctanG6#*&%\"xG\"\"\"%\"aG!\" \"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {TEXT 315 31 "_______________________________" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(1/((x^2+a^2)^2),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$),&*$)%\"xG\"\"#F'F'*$)%\"aGF.F'F'F.F'!\"\" F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**\"\"#!\"\"%\"xG\"\"\"%\"aG! \"#,&*$)F'F%F(F(*$)F)F%F(F(F&F(*&#F(F%F(*&F)!\"$-%'arctanG6#*&F'F(F)F& F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 71 " : This integral can also be found by making the (inverse) substitution " }{XPPEDIT 18 0 "x=a*tan*theta" "6#/%\"xG*(%\"aG\"\"\"%$tanGF'%&thet aGF'" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^2),x)" "6#-%$IntG6$*&\"\"\"F'*$,&*$%\"xG\"\"#F' *$%\"aGF,F'F,!\"\"F+" }{TEXT -1 7 " --- " }{XPPEDIT 18 0 "PIECEWISE( [x = a*tan*theta, theta = arctan(x/a)],[dx = a*sec^2*d*theta, ``]);" " 6#-%*PIECEWISEG6$7$/%\"xG*(%\"aG\"\"\"%$tanGF+%&thetaGF+/F--%'arctanG6 #*&F(F+F*!\"\"7$/%#dxG**F*F+*$%$secG\"\"#F+%\"dGF+F-F+%!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(a*sec^2*theta/((a^2*tan^2*theta+a^2)^2),th eta);" "6#/%!G-%$IntG6$**%\"aG\"\"\"*$%$secG\"\"#F*%&thetaGF**$,&*(F)F -%$tanGF-F.F*F**$F)F-F*F-!\"\"F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int(a*sec^2*theta/(a^4*sec^4*theta), theta)" "6#/%!G-%$IntG6$**%\"aG\"\"\"*$%$secG\"\"#F*%&thetaGF**(F)\"\" %F,F0F.F*!\"\"F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= 1/a^3" "6#/%!G*&\"\"\"F&*$%\"aG\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos^2*theta,theta)" "6#-%$IntG6$*&%$cosG \"\"#%&thetaG\"\"\"F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(2*a^3)" "6#/%!G*&\"\"\"F&*&\"\"#F&*$%\"aG\" \"$F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(cos*2*theta+1),thet a);" "6#-%$IntG6$-%!G6#,&*(%$cosG\"\"\"\"\"#F,%&thetaGF,F,F,F,F." } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1 /(2*a^3)" "6#/%!G*&\"\"\"F&*&\"\"#F&*$%\"aG\"\"$F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sin*2*theta/2+theta) + c" "6#,&-%!G6#,&**%$sinG \"\"\"\"\"#F*%&thetaGF*F+!\"\"F*F,F*F*%\"cGF*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=sin*theta*cos*theta/(2*a^3)+theta/(2*a^3) + c" "6#/% !G,(*,%$sinG\"\"\"%&thetaGF(%$cosGF(F)F(*&\"\"#F(*$%\"aG\"\"$F(!\"\"F( *&F)F(*&F,F(*$F.F/F(F0F(%\"cGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "theta=arctan(x/a)" "6#/%&thetaG-%' arctanG6#*&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 17 " it follows that " } {XPPEDIT 18 0 "sin*theta=x/sqrt(x^2+a^2)" "6#/*&%$sinG\"\"\"%&thetaGF& *&%\"xGF&-%%sqrtG6#,&*$F)\"\"#F&*$%\"aGF/F&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*theta=a/sqrt(x^2+a^2)" "6#/*&%$cosG\"\"\"%&thetaGF &*&%\"aGF&-%%sqrtG6#,&*$%\"xG\"\"#F&*$F)F0F&!\"\"" }{TEXT -1 9 " so th at " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^ 2)^2),x)=1/(2*a^3)" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF- F(F-!\"\"F,*&F(F(*&F-F(*$F/\"\"$F(F0" }{XPPEDIT 18 0 "``(x/sqrt(x^2+a^ 2))*`.`*``(a/sqrt(x^2+a^2))+1/(2*a^3);" "6#,&*(-%!G6#*&%\"xG\"\"\"-%%s qrtG6#,&*$F)\"\"#F**$%\"aGF0F*!\"\"F*%\".GF*-F&6#*&F2F*-F,6#,&*$F)F0F* *$F2F0F*F3F*F**&F*F**&F0F**$F2\"\"$F*F3F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c" "6#,&-%'arctanG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF) " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = x/(2*a^2*(x^2+a^2))+1/(2*a^3);" "6#/%!G,&*&%\"xG\"\"\"*(\"\"#F(*$% \"aGF*F(,&*$F'F*F(*$F,F*F(F(!\"\"F(*&F(F(*&F*F(*$F,\"\"$F(F0F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c" "6#,&-%'arctanG6#*&%\"xG \"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 229 "Int(1/((x^2+a^2)^2),x);\n``=student[changev ar](x=a*tan(theta),%,theta);\n``=simplify(convert(rhs(%),sincos));\n`` =map(combine,rhs(%));\n``=frontend(expand,[value(rhs(%))]);\n``=map(no rmal,expand(eval(subs(theta=arctan(x/a),rhs(%)))));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$),&*$)%\"xG\"\"#F'F'*$)%\"aGF.F' F'F.F'!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&\"\"\" F)*&)%\"aG\"\"$F),&F)F)*$)-%$tanG6#%&thetaG\"\"#F)F)F)!\"\"F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$-%$IntG6$*$)-%$cosG6#%&thetaG\" \"#\"\"\"F.*$%\"aG!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$-%$Int G6$,&*&#\"\"\"\"\"#F,-%$cosG6#,$*&F-F,%&thetaGF,F,F,F,F+F,F3*$%\"aG!\" $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"%F(*&%\"aG!\"$ -%$sinG6#,$*&\"\"#F(%&thetaGF(F(F(F(F(*(F2!\"\"F+F,F3F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&**\"\"#!\"\"%\"xG\"\"\"%\"aG!\"#,&*$)F)F' F*F**$)F+F'F*F*F(F**&#F*F'F**&F+!\"$-%'arctanG6#*&F)F*F+F(F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The special case where " } {XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+1)^2),x) = x/(2 *(x^2+1))+1/2;" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(F(F(F-!\"\"F, ,&*&F,F(*&F-F(,&*$F,F-F(F(F(F(F.F(*&F(F(F-F.F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "arctan*x+c;" "6#,&*&%'arctanG\"\"\"%\"xGF&F&%\"cGF&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x/(2*(x^2+1))+1/2;" "6#/-%\"f G6#%\"xG,&*&F'\"\"\"*&\"\"#F*,&*$F'F,F*F*F*F*!\"\"F**&F*F*F,F/F*" } {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x;" "6#*&%'arctanG\"\"\"%\"xGF% " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "g(x) = 1/((x^2+1)^2);" "6#/-% \"gG6#%\"xG*&\"\"\"F)*$,&*$F'\"\"#F)F)F)F-!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the fo llowing picture the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 45 "The picture is consistent with the fact that " } {XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "6#/-%%DiffG6$7#-%\"fG6#%\"xGF+ -%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = in finity) = Pi/4;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*%)infinityG*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Limit(f(x),x = -inf inity) = -Pi/4;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*,$%)infinityG!\"\",$*& %#PiG\"\"\"\"\"%F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "f := x -> x/(2*(x^2+1))+a rctan(x)/2:\n'f(x)'=f(x);\ng := x -> 1/((x^2+1)^2):\n'g(x)'=g(x);\np1 \+ := plot([f(x),g(x)],x=-2..2,-1..1.2,color=[red,blue],discont=true):\np 2 := plot([Pi/4,-Pi/4],x=-2..2,-1..1.2,color=COLOR(RGB,.4,.4,.4),lines tyle=3):\nplots[display]([p1,p2],tickmarks=[4,4]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&F'\"\"\",&*&\"\"#F*)F'F-F*F*F-F*!\" \"F**&#F*F-F*-%'arctanGF&F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\" gG6#%\"xG*&\"\"\"F)*$),&*$)F'\"\"#F)F)F)F)F/F)!\"\"" }}{PARA 13 "" 1 " " {GLPLOT2D 539 281 281 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$!\"#\"\"!$!3g^ /(*)eVd`(!#=7$$!3MLLL$Q6G\">!#<$!3]/p/rYH)\\(F-7$$!3bmm;M!\\p$=F1$!3uA v^eT0huF-7$$!3MLLL))Qj^'***F-$!3AR'[18SgU'F-7$$!3E++++0\" *H\"*F-$!3sgr2-#4%*='F-7$$!35++++83&H)F-$!32)z%e,YL>fF-7$$!3\\LLL3k(p` (F-$!3]G;p?ufKcF-7$$!3Anmmmj^NmF-$!3L]$3NJKFB&F-7$$!3)zmmmYh=(eF-$!3m& ymGF9z$[F-7$$!3+,++v#\\N)\\F-$!3\"ob;>s&p2VF-7$$!3commmCC(>%F-$!3+F)yZ 6a7x$F-7$$!39*****\\FRXL$F-$!3e]509&H(4JF-7$$!3t*****\\#=/8DF-$!3CE>vD %4HT#F-7$$!3=mmm;a*el\"F-$!3_R/#Rv`ji\"F-7$$!3komm;Wn(o)!#>$!3l8?Qe`DW ')Fjr7$$!3IqLLL$eV(>!#?$!3))HE`-KNu>F`s7$$\"3)Qjmm\"f`@')Fjr$\"3;LP&\\ i'4z&)Fjr7$$\"3%z****\\nZ)H;F-$\"3;e#\\D;d;g\"F-7$$\"3ckmm;$y*eCF-$\"3 /@J))fk&\\O#F-7$$\"3f)******R^bJ$F-$\"3YE+y\\SL%4$F-7$$\"3'e*****\\5a` TF-$\"3-c+C@^cRPF-7$$\"3'o****\\7RV'\\F-$\"3!yesN%*\\`H%F-7$$\"3Y'**** *\\@fkeF-$\"3:yjK*3#*Q$[F-7$$\"3_ILLL&4Nn'F-$\"3;;!f:(H)4D&F-7$$\"3A** *****\\,s`(F-$\"3)z/wN'*)oKcF-7$$\"3%[mm;zM)>$)F-$\"397wNt++GfF-7$$\"3 M*******pfa<*F-$\"3s$)\\u1l*G?'F-7$$\"39HLLeg`!)**F-$\"3;\"GY\"[`6AkF- 7$$\"3w****\\#G2A3\"F1$\"3WHH(F-7$$\"3ILLLGUYo;F1$\"3-7JVFS -ftF-7$$\"3_mmm1^rZF1$\"3/adMuAU)\\(F-7$$\"\"#F*$\"3g^/(*)eVd`(F--%'COLO URG6&%$RGBG$\"*++++\"!\")$F*F*Fb[l-F$6$7gn7$F($\"3y+++++++SFjr7$F/$\"3 ,Cu!*)Gjsg%Fjr7$F5$\"3>C;?*omfA&Fjr7$F:$\"3unRrS69UgFjr7$F?$\"3Uf*G:\" fA=qFjr7$FD$\"3y&R2zkMf<)Fjr7$FI$\"35l(e$3TN\\%*Fjr7$FN$\"3$fIq$ei;,6F -7$FS$\"3!p6h[kISH\"F-7$FX$\"3)*R*HoHuU_\"F-7$Fgn$\"3\\_%pQ=)\\3=F-7$F \\o$\"3Ex1#p\\4e5#F-7$Fao$\"3c\"RJEP,>]#F-7$Ffo$\"3ol7j$)[\\uHF-7$F[p$ \"3n`$)oq:B4NF-7$F`p$\"3i4eW^P+nSF-7$Fep$\"3mR\"='>p^?[F-7$Fjp$\"3scvp GheHbF-7$F_q$\"38:mmL5&oT'F-7$Fdq$\"3\"\\*Rx#)) 4)F-7$F^r$\"3CmpdE,CZ))F-7$Fcr$\"3!y;.8LgLZ*F-7$$!3_mm;H9Li7F-$\"31\\) 4?ti()o*F-7$Fhr$\"3k))p\"HcS2&)*F-7$$!3sNL$3x9^c'Fjr$\"3^:K]_EN9**F-7$ $!3$G++]7bDW%Fjr$\"3+TbedRkg**F-7$$!3$*pm;za**>BFjr$\"3Yq:sERC*)**F-7$ F^s$\"3'RGR'Q?#*****F-7$$\"3O$fmTNc$\\!*F`s$\"3C\")G:%Qi$)***F-7$$\"3q bm;/rI2?Fjr$\"31\"Q(H/j%>***F-7$$\"31_m\"Hdy'4JFjr$\"3i#eua\"yo!)**F-7 $$\"3V[mmT+07UFjr$\"3QAZ$GZ6Y'**F-7$$\"3:Tm;zHz;kFjr$\"3&)>'4wPb\"=**F -7$Fds$\"33s;y%[zH&)*F-7$$\"3mILLL1+Y7F-$\"3'e&p['e!e'p*F-7$Fis$\"3UH# z>3j\"*[*F-7$F^t$\"3A?fA0'[@*))F-7$Fct$\"3mOlI!\\ls6)F-7$Fht$\"3%zaHJh !ytsF-7$F]u$\"3!H<9!eAaOkF-7$Fbu$\"3O41&*p*3m`&F-7$Fgu$\"3#=gni#\\$F-7$Ffv$\"39Xs%\\pDw%HF-7$F [w$\"34*45:$4v4DF-7$F`w$\"3P)*GPw\\M@@F-7$Few$\"3?[3w:<`,=F-7$Fjw$\"3r E.\\)Ro5_\"F-7$F_x$\"3mXx'oN'R&H\"F-7$Fdx$\"3mi,)oU_B5\"F-7$Fix$\"3#\\ rJJXeIU*Fjr7$F^y$\"3av#>S*4>#=)Fjr7$Fcy$\"3i$yv*GTt%)pFjr7$Fhy$\"3AV`% **z\"3$3'Fjr7$F]z$\"3cS;j(=NxE&Fjr7$Fbz$\"3GN=#4un^g%Fjr7$FgzFg[l-F\\[ l6&F^[lFb[lFb[lF_[l-F$6%7S7$F($\"3!G[uRj\")R&yF-7$F/F_hl7$F5F_hl7$F:F_ hl7$F?F_hl7$FDF_hl7$FIF_hl7$FNF_hl7$FSF_hl7$FXF_hl7$FgnF_hl7$F\\oF_hl7 $FaoF_hl7$FfoF_hl7$F[pF_hl7$F`pF_hl7$FepF_hl7$FjpF_hl7$F_qF_hl7$FdqF_h l7$FiqF_hl7$F^rF_hl7$FcrF_hl7$FhrF_hl7$F^sF_hl7$FdsF_hl7$FisF_hl7$F^tF _hl7$FctF_hl7$FhtF_hl7$F]uF_hl7$FbuF_hl7$FguF_hl7$F\\vF_hl7$FavF_hl7$F fvF_hl7$F[wF_hl7$F`wF_hl7$FewF_hl7$FjwF_hl7$F_xF_hl7$FdxF_hl7$FixF_hl7 $F^yF_hl7$FcyF_hl7$FhyF_hl7$F]zF_hl7$FbzF_hl7$FgzF_hl-%&COLORG6&F^[l$ \"\"%!\"\"Fd[mFd[m-%*LINESTYLEG6#\"\"$-F$6%7S7$F($!3!G[uRj\")R&yF-7$F/ F_\\m7$F5F_\\m7$F:F_\\m7$F?F_\\m7$FDF_\\m7$FIF_\\m7$FNF_\\m7$FSF_\\m7$ FXF_\\m7$FgnF_\\m7$F\\oF_\\m7$FaoF_\\m7$FfoF_\\m7$F[pF_\\m7$F`pF_\\m7$ FepF_\\m7$FjpF_\\m7$F_qF_\\m7$FdqF_\\m7$FiqF_\\m7$F^rF_\\m7$FcrF_\\m7$ FhrF_\\m7$F^sF_\\m7$FdsF_\\m7$FisF_\\m7$F^tF_\\m7$FctF_\\m7$FhtF_\\m7$ F]uF_\\m7$FbuF_\\m7$FguF_\\m7$F\\vF_\\m7$FavF_\\m7$FfvF_\\m7$F[wF_\\m7 $F`wF_\\m7$FewF_\\m7$FjwF_\\m7$F_xF_\\m7$FdxF_\\m7$FixF_\\m7$F^yF_\\m7 $FcyF_\\m7$FhyF_\\m7$F]zF_\\m7$FbzF_\\m7$FgzF_\\mFa[mFg[m-%*AXESTICKSG 6$Fe[mFe[m-%+AXESLABELSG6%Q\"x6\"Q!Fh_m-%%FONTG6#%(DEFAULTG-%%VIEWG6$; F(Fgz;$Ff[mF*$\"#7Ff[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 258 "" 0 "" {XPPEDIT 18 0 "Int(1/((x^2+a^2)^3),x) = x/(4 *a^2*(x^2+a^2)^2)+3*x/(8*a^4*(x^2+a^2))+3/(8*a^5);" "6#/-%$IntG6$*&\" \"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF-F(\"\"$!\"\"F,,(*&F,F(*(\"\"%F(*$F/F- F(,&*$F,F-F(*$F/F-F(F-F1F(*(F0F(F,F(*(\"\")F(*$F/F5F(,&*$F,F-F(*$F/F-F (F(F1F(*&F0F(*&F " 0 "" {MPLTEXT 1 0 161 "unassign ('x','A','B','C');\neq := 1/((x^2+a^2)^3)=A*(a^2-3*x^2)/((x^2+a^2)^3)+ B*(a^2-x^2)/((x^2+a^2)^2)+C/(x^2+a^2);\nsolve(identity(eq,x),\{A,B,C\} );\nassign(%);\neq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/*&\"\"\" F'*$),&*$)%\"xG\"\"#F'F'*$)%\"aGF.F'F'\"\"$F'!\"\",(*(%\"AGF',&*&F2F'F ,F'F3F/F'F'F*!\"$F'*(%\"BGF',&F/F'F+F3F'F*!\"#F'*&%\"CGF'F*F3F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"CG,$*(\"\"$\"\"\"\"\")!\"\"%\"aG !\"%F)/%\"BGF&/%\"AG,$*&F)F)*&\"\"%F))F,\"\"#F)F+F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&\"\"\"F%*$),&*$)%\"xG\"\"#F%F%*$)%\"aGF,F%F%\"\"$F %!\"\",(**\"\"%F1F/!\"#,&*&F0F%F*F%F1F-F%F%F(!\"$F%*,F0F%\"\")F1F/!\"% ,&F-F%F)F1F%F(F5F%**F0F%F:F1F/F;F(F1F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The expansion (iii) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/((x^2+a^2)^3) = (a^2-3*x^2)/(4*a^2*(x^2+a^2 )^3)+3*(a^2-x^2)/(8*a^4*(x^2+a^2)^2)+3/(8*a^4*(x^2+a^2));" "6#/*&\"\" \"F%*$,&*$%\"xG\"\"#F%*$%\"aGF*F%\"\"$!\"\",(*&,&*$F,F*F%*&F-F%*$F)F*F %F.F%*(\"\"%F%*$F,F*F%,&*$F)F*F%*$F,F*F%F-F.F%*(F-F%,&*$F,F*F%*$F)F*F. F%*(\"\")F%*$F,F6F%,&*$F)F*F%*$F,F*F%F*F.F%*&F-F%*(F@F%*$F,F6F%,&*$F)F *F%*$F,F*F%F%F.F%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so t hat " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a ^2)^3),x) = 1/(4*a^4)" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"a GF-F(\"\"$!\"\"F,*&F(F(*&\"\"%F(*$F/F4F(F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((a^2-3*x^2)/(x^2+a^2)^3,x) +3/(8*a^4)" "6#,&-%$IntG6$*&,&*$% \"aG\"\"#\"\"\"*&\"\"$F,*$%\"xGF+F,!\"\"F,*$,&*$F0F+F,*$F*F+F,F.F1F0F, *&F.F,*&\"\")F,*$F*\"\"%F,F1F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((a ^2-x^2)/(x^2+a^2)^2,x) +3/(8*a^4)" "6#,&-%$IntG6$*&,&*$%\"aG\"\"#\"\" \"*$%\"xGF+!\"\"F,*$,&*$F.F+F,*$F*F+F,F+F/F.F,*&\"\"$F,*&\"\")F,*$F*\" \"%F,F/F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^2+a^2),x)" "6#-%$I ntG6$*&\"\"\"F',&*$%\"xG\"\"#F'*$%\"aGF+F'!\"\"F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 31 "Using (i) and (ii) we see that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^3),x) =x/( 4*a^2*(x^2+a^2)^2)+3*x/(8*a^4*(x^2+a^2))+3/(8*a^5)" "6#/-%$IntG6$*&\" \"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF-F(\"\"$!\"\"F,,(*&F,F(*(\"\"%F(*$F/F- F(,&*$F,F-F(*$F/F-F(F-F1F(*(F0F(F,F(*(\"\")F(*$F/F5F(,&*$F,F-F(*$F/F-F (F(F1F(*&F0F(*&F " 0 "" {MPLTEXT 1 0 33 "Int(1/( (x^2+a^2)^3),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*&\"\"\"F'*$),&*$)%\"xG\"\"#F'F'*$)%\"aGF.F'F'\"\"$F'!\"\"F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(**\"\"%!\"\"%\"xG\"\"\"%\"aG!\"#,&*$ )F'\"\"#F(F(*$)F)F.F(F(F*F(*,\"\"$F(\"\")F&F)!\"%F'F(F+F&F(*&#F2F3F(*& F)!\"&-%'arctanG6#*&F'F(F)F&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 71 ": This integral can also be found by \+ making the (inverse) substitution " }{XPPEDIT 18 0 "x=a*tan*theta" "6# /%\"xG*(%\"aG\"\"\"%$tanGF'%&thetaGF'" }{TEXT -1 2 ". " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^3),x);" "6#-%$I ntG6$*&\"\"\"F'*$,&*$%\"xG\"\"#F'*$%\"aGF,F'\"\"$!\"\"F+" }{TEXT -1 7 " --- " }{XPPEDIT 18 0 "PIECEWISE([x = a*tan*theta, theta = arctan(x /a)],[dx = a*sec^2*d*theta, ``]);" "6#-%*PIECEWISEG6$7$/%\"xG*(%\"aG\" \"\"%$tanGF+%&thetaGF+/F--%'arctanG6#*&F(F+F*!\"\"7$/%#dxG**F*F+*$%$se cG\"\"#F+%\"dGF+F-F+%!G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int(a*sec^2 *theta/((a^2*tan^2*theta+a^2)^3),theta);" "6#/%!G-%$IntG6$**%\"aG\"\" \"*$%$secG\"\"#F*%&thetaGF**$,&*(F)F-%$tanGF-F.F*F**$F)F-F*\"\"$!\"\"F ." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = Int(a*sec^2*theta/(a^6*sec^6*theta),theta);" "6#/%!G-%$IntG6$**% \"aG\"\"\"*$%$secG\"\"#F*%&thetaGF**(F)\"\"'F,F0F.F*!\"\"F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(a^ 5);" "6#/%!G*&\"\"\"F&*$%\"aG\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(cos^4*theta,theta);" "6#-%$IntG6$*&%$cosG\"\"%%&thetaG\"\"\"F) " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = 1/(a^5);" "6#/%!G*&\"\"\"F&*$%\"aG\"\"&!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int((cos*2*theta+1)^2/4,theta);" "6#-%$IntG6$*&,&*(%$co sG\"\"\"\"\"#F*%&thetaGF*F*F*F*F+\"\"%!\"\"F," }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(4*a^5);" "6#/ %!G*&\"\"\"F&*&\"\"%F&*$%\"aG\"\"&F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(cos^2*2*theta+2*cos*2*theta+1),theta);" "6#-%$IntG6$-%!G6 #,(*(%$cosG\"\"#F,\"\"\"%&thetaGF-F-**F,F-F+F-F,F-F.F-F-F-F-F." } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(4*a^3);" "6#/%!G*&\"\"\"F&*&\"\"%F&*$%\"aG\"\"$F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``((cos*4*theta+1)/2+2*cos*2*theta+1),theta) ;" "6#-%$IntG6$-%!G6#,(*&,&*(%$cosG\"\"\"\"\"%F.%&thetaGF.F.F.F.F.\"\" #!\"\"F.**F1F.F-F.F1F.F0F.F.F.F.F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(8*a^5);" "6#/%!G*&\"\"\"F&*& \"\")F&*$%\"aG\"\"&F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(cos *4*theta+4*cos*2*theta+3),theta);" "6#-%$IntG6$-%!G6#,(*(%$cosG\"\"\" \"\"%F,%&thetaGF,F,**F-F,F+F,\"\"#F,F.F,F,\"\"$F,F." }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(8*a^5);" "6# /%!G*&\"\"\"F&*&\"\")F&*$%\"aG\"\"&F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(sin*4*theta/4+2*sin*2*theta+3*theta)+c;" "6#,&-%!G6#,(**%$sin G\"\"\"\"\"%F*%&thetaGF*F+!\"\"F***\"\"#F*F)F*F/F*F,F*F**&\"\"$F*F,F*F *F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sin*4*theta/(32*a^5)+sin*2*theta/(4*a^5)+3*theta/( 8*a^5)+c;" "6#/%!G,***%$sinG\"\"\"\"\"%F(%&thetaGF(*&\"#KF(*$%\"aG\"\" &F(!\"\"F(**F'F(\"\"#F(F*F(*&F)F(*$F.F/F(F0F(*(\"\"$F(F*F(*&\"\")F(*$F .F/F(F0F(%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin*2*theta*cos* 2*theta/(16*a^5)+sin*theta*cos*theta/(2*a^5)+3*theta/(8*a^5)+c;" "6#/% !G,**0%$sinG\"\"\"\"\"#F(%&thetaGF(%$cosGF(F)F(F*F(*&\"#;F(*$%\"aG\"\" &F(!\"\"F(*,F'F(F*F(F+F(F*F(*&F)F(*$F/F0F(F1F(*(\"\"$F(F*F(*&\"\")F(*$ F/F0F(F1F(%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin*theta*cos*th eta*(2*cos^2*theta-1)/(8*a^5)+sin*theta*cos*theta/(2*a^5)+3*theta/(8*a ^5)+c;" "6#/%!G,**.%$sinG\"\"\"%&thetaGF(%$cosGF(F)F(,&*(\"\"#F(*$F*F- F(F)F(F(F(!\"\"F(*&\"\")F(*$%\"aG\"\"&F(F/F(*,F'F(F)F(F*F(F)F(*&F-F(*$ F3F4F(F/F(*(\"\"$F(F)F(*&F1F(*$F3F4F(F/F(%\"cGF(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = sin*theta*cos^3*theta/(4*a^5)+3*sin*theta*cos*thet a/(8*a^5)+3*theta/(8*a^5)+c" "6#/%!G,**,%$sinG\"\"\"%&thetaGF(%$cosG\" \"$F)F(*&\"\"%F(*$%\"aG\"\"&F(!\"\"F(*.F+F(F'F(F)F(F*F(F)F(*&\"\")F(*$ F/F0F(F1F(*(F+F(F)F(*&F4F(*$F/F0F(F1F(%\"cGF(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "theta=arctan(x/a)" "6#/%&thetaG-%'arctanG6#*&%\"xG\"\" \"%\"aG!\"\"" }{TEXT -1 17 " it follows that " }{XPPEDIT 18 0 "sin*the ta=x/sqrt(x^2+a^2)" "6#/*&%$sinG\"\"\"%&thetaGF&*&%\"xGF&-%%sqrtG6#,&* $F)\"\"#F&*$%\"aGF/F&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos*th eta=a/sqrt(x^2+a^2)" "6#/*&%$cosG\"\"\"%&thetaGF&*&%\"aGF&-%%sqrtG6#,& *$%\"xG\"\"#F&*$F)F0F&!\"\"" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^3),x) = x/(4*a^2*(x ^2+a^2)^2)+3*x/(8*a^4*(x^2+a^2))+3/(8*a^5);" "6#/-%$IntG6$*&\"\"\"F(*$ ,&*$%\"xG\"\"#F(*$%\"aGF-F(\"\"$!\"\"F,,(*&F,F(*(\"\"%F(*$F/F-F(,&*$F, F-F(*$F/F-F(F-F1F(*(F0F(F,F(*(\"\")F(*$F/F5F(,&*$F,F-F(*$F/F-F(F(F1F(* &F0F(*&F " 0 "" {MPLTEXT 1 0 229 "Int(1/((x^2+a^2)^ 3),x);\n``=student[changevar](x=a*tan(theta),%,theta);\n``=simplify(co nvert(rhs(%),sincos));\n``=map(combine,rhs(%));\n``=frontend(expand,[v alue(rhs(%))]);\n``=map(normal,expand(eval(subs(theta=arctan(x/a),rhs( %)))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$),&*$) %\"xG\"\"#F'F'*$)%\"aGF.F'F'\"\"$F'!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&\"\"\"F)*&)%\"aG\"\"&F)),&F)F)*$)-%$tanG 6#%&thetaG\"\"#F)F)F6F)!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G ,$-%$IntG6$*$)-%$cosG6#%&thetaG\"\"%\"\"\"F.*$%\"aG!\"&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%!G,$-%$IntG6$,(*&#\"\"\"\"\")F,-%$cosG6#,$*&\" \"%F,%&thetaGF,F,F,F,*&#F,\"\"#F,-F/6#,$*&F7F,F4F,F,F,F,#\"\"$F-F,F4*$ %\"aG!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"\"\"#KF(*&% \"aG!\"&-%$sinG6#,$*&\"\"%F(%&thetaGF(F(F(F(F(*&#F(F2F(*&F+F,-F.6#,$*& \"\"#F(F3F(F(F(F(F(**\"\"$F(\"\")!\"\"F+F,F3F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(**\"\"%!\"\"%\"xG\"\"\"%\"aG!\"#,&*$)F)\"\"#F*F** $)F+F0F*F*F,F**,\"\"$F*\"\")F(F+!\"%F)F*F-F(F**&#F4F5F**&F+!\"&-%'arct anG6#*&F)F*F+F(F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The sp ecial case where " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x ^2+1)^3),x) = x/(4*(x^2+1)^2)+3*x/(8*(x^2+1))+3/8;" "6#/-%$IntG6$*&\" \"\"F(*$,&*$%\"xG\"\"#F(F(F(\"\"$!\"\"F,,(*&F,F(*&\"\"%F(*$,&*$F,F-F(F (F(F-F(F/F(*(F.F(F,F(*&\"\")F(,&*$F,F-F(F(F(F(F/F(*&F.F(F9F/F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x+c;" "6#,&*&%'arctanG\"\"\"%\"x GF&F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = x/(4*(x^2+1 )^2)+3*x/(8*(x^2+1))+3/8;" "6#/-%\"fG6#%\"xG,(*&F'\"\"\"*&\"\"%F**$,&* $F'\"\"#F*F*F*F0F*!\"\"F**(\"\"$F*F'F**&\"\")F*,&*$F'F0F*F*F*F*F1F**&F 3F*F5F1F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x;" "6#*&%'arctanG\" \"\"%\"xGF%" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "g(x) = 1/((x^2+1)^3 );" "6#/-%\"gG6#%\"xG*&\"\"\"F)*$,&*$F'\"\"#F)F)F)\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph of " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 45 "The picture is consistent with the fact t hat " }{XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "6#/-%%DiffG6$7#-%\"fG6# %\"xGF+-%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Not e that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x) ,x = infinity) = 3*Pi/16;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*%)infinityG* (\"\"$\"\"\"%#PiGF/\"#;!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Lim it(f(x),x = -infinity) = -3*Pi/16;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*,$% )infinityG!\"\",$*(\"\"$\"\"\"%#PiGF2\"#;F.F." }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 315 "f := x -> x/(4*(x^2+1)^2)+3*x/(8*(x^2+1))+3/8*arctan(x):\n'f(x)'= f(x);\ng := x -> 1/((x^2+1)^3):\n'g(x)'=g(x);\np1 := plot([f(x),g(x)], x=-1.6..1.6,-.8..1.2,color=[red,blue],discont=true):\np2 := plot([3*Pi /16,-3*Pi/16],x=-1.6..1.6,-.8..1.2,color=COLOR(RGB,.4,.4,.4),linestyle =3):\nplots[display]([p1,p2],tickmarks=[4,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*(\"\"%!\"\"F'\"\"\",&*$)F'\"\"#F,F,F,F ,!\"#F,*(\"\"$F,F'F,,&*&\"\")F,F/F,F,F6F,F+F,*&#F3F6F,-%'arctanGF&F,F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"\"\"F)*$),&*$)F' \"\"#F)F)F)F)\"\"$F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 498 260 260 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$!33+++++++;!#<$!3O*pIkw[nz&!#=7$$!3!om mm5\\-`\"F*$!3o*o#eh*G(zdF-7$$!3MLLLF#f&p9F*$!3)=rb@%e&F-7$$!3!pmmEN+t1\"F*$!3]a6J%\\A9_&F-7$$!3+,++g+J'***F-$!3,ME& [h\"yWaF-7$$!3OLLL`wC+$*F-$!3sVL/p+#zM&F-7$$!3Cnmm1b:(o)F-$!3CnW,wT>W_ F-7$$!3;+++g*ep*zF-$!3)eM%\\&3iC5&F-7$$!3a,+++%GRI(F-$!3$>)HNN]DG\\F-7 $$!3(4+++/lgj'F-$!3\"Gf,'G$)RCZF-7$$!3onmmE6eHgF-$!3k&4m0%[o.XF-7$$!36 NLL$48%3`F-$!3A$zqr&o\"3>%F-7$$!3]LLLt\"*[(p%F-$!3cR/!RH0\"yQF-7$$!3C, ++?%Ro)RF-$!3g*fIqNAZX$F-7$$!3=OLLtRzdLF-$!3g&He16dV-$F-7$$!3k+++?9jnE F-$!3WeAB&)*QF\\#F-7$$!3n+++gMV5?F-$!3IjtC?]$H$>F-7$$!3QLLLLjrC8F-$!3N e=ml([>I\"F-7$$!3yVLLL&R,&p!#>$!3xA\\vj2w;pFir7$$!3mSnmmm[z:!#?$!3eW(4 is#[z:F_s7$$\"3)fJLLtGs*oFir$\"3q'H#o^OgkoFir7$$\"3!))*****R\"yQI\"F-$ \"3)eQqM>a@G\"F-7$$\"3aKLL`E=n>F-$\"3s?gBudV%*=F-7$$\"3a(*****>6W_EF-$ 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\\,x<;-\"F-7$FW$\"3!o&>j$\\%Q^7F-7$Ffn$\"3r1U#=h/+#H7#)Gk8)F-7$F]r$\"3Kh \"38-U#z))F-7$Fbr$\"3SB`$3$*)[\"\\*F-7$$!3)QLLL9l)45F-$\"3+jXZSv=+(*F- 7$Fgr$\"3U'ezG[vk&)*F-7$$!3Owmm;=4__Fir$\"3<:(Gh/,x\"**F-7$$!3%*3+++T/ aNFir$\"3E4HBR=?i**F-7$$!3]TLL$Q'*f&=Fir$\"3Q**y&y%Hn*)**F-7$F]s$\"35J &Hq:D*****F-7$$\"3XBLL$oXeg\"Fir$\"3[()3:nxE#***F-7$$\"3'4KLL.S'pLFir$ \"3-gpW'y8g'**F-7$$\"3[=LL$QMM8&Fir$\"3sxh&HRe8#**F-7$Fcs$\"3KRu_h=je) *F-7$$\"3)>lmm10!o**Fir$\"3g.%Q7bVxq*F-7$Fhs$\"3z&H&['RLo]*F-7$F]t$\"3 bS$[m0MM#*)F-7$Fbt$\"3W8x!o9s[:)F-7$Fgt$\"3ao[NqVx.tF-7$F\\u$\"3b>jDP#>4Z0bF-7$Ffu$\"31?5*H`5Er%F-7$F[v$\"3]9!zm,\">WRF-7 $F`v$\"3_Z*zvl&4GLF-7$Fev$\"3kp\"e&H[QWFF-7$Fjv$\"3i?fV8=WxAF-7$F_w$\" 3%HsWK'yKn=F-7$Fdw$\"3J4dqPw^M:F-7$Fiw$\"3/c'))f$z9[7F-7$F^x$\"3vY%fw, jH-\"F-7$Fcx$\"3k&)\\!*>X>`$)Fir7$Fhx$\"3xChn)4yD%oFir7$F]y$\"3_peEPvi 1dFir7$Fby$\"3KtE+,*>jk%Fir7$Fgy$\"3GjScQ+&f(QFir7$F\\z$\"36AiPH_-/KFi r7$Faz$\"3qvL\\'**H#yEFir7$FfzFg[l-F[[l6&F][lFa[lFa[lF^[l-F$6%7S7$F($ \"33i3[Di[!*eF-7$F/Fegl7$F4Fegl7$F9Fegl7$F>Fegl7$FCFegl7$FHFegl7$FMFeg l7$FRFegl7$FWFegl7$FfnFegl7$F[oFegl7$F`oFegl7$FeoFegl7$FjoFegl7$F_pFeg l7$FdpFegl7$FipFegl7$F^qFegl7$FcqFegl7$FhqFegl7$F]rFegl7$FbrFegl7$FgrF egl7$F]sFegl7$FcsFegl7$FhsFegl7$F]tFegl7$FbtFegl7$FgtFegl7$F\\uFegl7$F auFegl7$FfuFegl7$F[vFegl7$F`vFegl7$FevFegl7$FjvFegl7$F_wFegl7$FdwFegl7 $FiwFegl7$F^xFegl7$FcxFegl7$FhxFegl7$F]yFegl7$FbyFegl7$FgyFegl7$F\\zFe gl7$FazFegl7$FfzFegl-%&COLORG6&F][l$\"\"%!\"\"FjjlFjjl-%*LINESTYLEG6# \"\"$-F$6%7S7$F($!33i3[Di[!*eF-7$F/Fe[m7$F4Fe[m7$F9Fe[m7$F>Fe[m7$FCFe[ m7$FHFe[m7$FMFe[m7$FRFe[m7$FWFe[m7$FfnFe[m7$F[oFe[m7$F`oFe[m7$FeoFe[m7 $FjoFe[m7$F_pFe[m7$FdpFe[m7$FipFe[m7$F^qFe[m7$FcqFe[m7$FhqFe[m7$F]rFe[ m7$FbrFe[m7$FgrFe[m7$F]sFe[m7$FcsFe[m7$FhsFe[m7$F]tFe[m7$FbtFe[m7$FgtF e[m7$F\\uFe[m7$FauFe[m7$FfuFe[m7$F[vFe[m7$F`vFe[m7$FevFe[m7$FjvFe[m7$F _wFe[m7$FdwFe[m7$FiwFe[m7$F^xFe[m7$FcxFe[m7$FhxFe[m7$F]yFe[m7$FbyFe[m7 $FgyFe[m7$F\\zFe[m7$FazFe[m7$FfzFe[mFgjlF][m-%*AXESTICKSG6$F[[mF[[m-%+ AXESLABELSG6%Q\"x6\"Q!F^_m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#;F\\[m$\"# ;F\\[m;$F`[lF\\[m$\"#7F\\[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (1/((x^2+a^2)^n),x) = x/((2*n-2)*a^2*(x^2+a^2)^(n-1))+(2*n-3)/((2*n-2) *a^2);" "6#/-%$IntG6$*&\"\"\"F(),&*$%\"xG\"\"#F(*$%\"aGF-F(%\"nG!\"\"F ,,&*&F,F(*(,&*&F-F(F0F(F(F-F1F(*$F/F-F(),&*$F,F-F(*$F/F-F(,&F0F(F(F1F( F1F(*&,&*&F-F(F0F(F(\"\"$F1F(*&,&*&F-F(F0F(F(F-F1F(*$F/F-F(F1F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^(n-1)),x);" "6#-%$IntG 6$*&\"\"\"F'),&*$%\"xG\"\"#F'*$%\"aGF,F',&%\"nGF'F'!\"\"F1F+" }{TEXT -1 8 ", where " }{TEXT 323 1 "n" }{TEXT -1 6 " > 1 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "First note that" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1/(x^2+a^2)^(n-1)]= d/d x" "6#/7#*&\"\"\"F&),&*$%\"xG\"\"#F&*$%\"aGF+F&,&%\"nGF&F&!\"\"F0*&%\" dGF&%#dxGF0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ (x^2+a^2)^(-n+1)]" "6#7 #),&*$%\"xG\"\"#\"\"\"*$%\"aGF(F),&%\"nG!\"\"F)F)" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-(n-1)*(x^2+a^2)^( -n)*`.`*2*x" "6#/%!G,$*,,&%\"nG\"\"\"F)!\"\"F)),&*$%\"xG\"\"#F)*$%\"aG F/F),$F(F*F)%\".GF)F/F)F.F)F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ -(2*n-2)*x/((x^2+a^2)^n);" "6#/%!G,$*(,&*&\"\"#\"\"\"%\"nGF*F*F)!\"\"F *%\"xGF*),&*$F-F)F**$%\"aGF)F*F+F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1/((2*n-2)*(x^2+a^2)^(n-1))] = -x/((x^2+a^2)^n);" "6#/7#*&\"\"\"F& *&,&*&\"\"#F&%\"nGF&F&F*!\"\"F&),&*$%\"xGF*F&*$%\"aGF*F&,&F+F&F&F,F&F, ,$*&F0F&),&*$F0F*F&*$F2F*F&F+F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We apply the integration \+ by parts formula to the integral " }{XPPEDIT 18 0 "Int(-x^2/((x^2+a^2) ^n),x);" "6#-%$IntG6$,$*&%\"xG\"\"#),&*$F(F)\"\"\"*$%\"aGF)F-%\"nG!\" \"F1F(" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(-x^2/((x^2+a^2)^n),x);" "6#-%$IntG6$,$*&%\"xG\"\"#) ,&*$F(F)\"\"\"*$%\"aGF)F-%\"nG!\"\"F1F(" }{TEXT -1 7 " --- " } {XPPEDIT 18 0 "PIECEWISE([u = x, v = 1/((2*n-2)*(x^2+a^2)^(n-1))],[du/ dx = 1, dv/dx = -x/((x^2+a^2)^n)]);" "6#-%*PIECEWISEG6$7$/%\"uG%\"xG/% \"vG*&\"\"\"F-*&,&*&\"\"#F-%\"nGF-F-F1!\"\"F-),&*$F)F1F-*$%\"aGF1F-,&F 2F-F-F3F-F37$/*&%#duGF-%#dxGF3F-/*&%#dvGF-F>F3,$*&F)F-),&*$F)F1F-*$F8F 1F-F2F3F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "The inegrati on by parts formula " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v-Int(v*``( du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG, &*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 8 " giv es: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(-x^2/((x^ 2+a^2)^n),x) = x/((2*n-2)*(x^2+a^2)^(n-1))-1/(2*n-2);" "6#/-%$IntG6$,$ *&%\"xG\"\"#),&*$F)F*\"\"\"*$%\"aGF*F.%\"nG!\"\"F2F),&*&F)F.*&,&*&F*F. F1F.F.F*F2F.),&*$F)F*F.*$F0F*F.,&F1F.F.F2F.F2F.*&F.F.,&*&F*F.F1F.F.F*F 2F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^2+a^2)^(n-1),x)" "6#-% $IntG6$*&\"\"\"F'),&*$%\"xG\"\"#F'*$%\"aGF,F',&%\"nGF'F'!\"\"F1F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(a^2/((x^2+a^2)^n),x)-Int((x^2+ a^2)/((x^2+a^2)^n),x) = x/((2*n-2)*(x^2+a^2)^(n-1))-1/(2*n-2);" "6#/,& -%$IntG6$*&%\"aG\"\"#),&*$%\"xGF*\"\"\"*$F)F*F/%\"nG!\"\"F.F/-F&6$*&,& *$F.F*F/*$F)F*F/F/),&*$F.F*F/*$F)F*F/F1F2F.F2,&*&F.F/*&,&*&F*F/F1F/F/F *F2F/),&*$F.F*F/*$F)F*F/,&F1F/F/F2F/F2F/*&F/F/,&*&F*F/F1F/F/F*F2F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^(n-1)),x)" "6#-%$IntG 6$*&\"\"\"F'),&*$%\"xG\"\"#F'*$%\"aGF,F',&%\"nGF'F'!\"\"F1F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(a^2/((x^2+a^2)^n),x)-Int(1/((x^2+ a^2)^(n-1)),x) = x/((2*n-2)*(x^2+a^2)^(n-1))-1/(2*n-2);" "6#/,&-%$IntG 6$*&%\"aG\"\"#),&*$%\"xGF*\"\"\"*$F)F*F/%\"nG!\"\"F.F/-F&6$*&F/F/),&*$ F.F*F/*$F)F*F/,&F1F/F/F2F2F.F2,&*&F.F/*&,&*&F*F/F1F/F/F*F2F/),&*$F.F*F /*$F)F*F/,&F1F/F/F2F/F2F/*&F/F/,&*&F*F/F1F/F/F*F2F2F2" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^(n-1)),x)" "6#-%$IntG6$*&\"\"\"F'), &*$%\"xG\"\"#F'*$%\"aGF,F',&%\"nGF'F'!\"\"F1F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(a^2/((x^2+a^2)^n),x) = x/((2*n-2)*(x^2+a^2)^(n-1 ))+(2*n-3)/(2*n-2);" "6#/-%$IntG6$*&%\"aG\"\"#),&*$%\"xGF)\"\"\"*$F(F) F.%\"nG!\"\"F-,&*&F-F.*&,&*&F)F.F0F.F.F)F1F.),&*$F-F)F.*$F(F)F.,&F0F.F .F1F.F1F.*&,&*&F)F.F0F.F.\"\"$F1F.,&*&F)F.F0F.F.F)F1F1F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^(n-1)),x)" "6#-%$IntG6$*&\"\"\"F' ),&*$%\"xG\"\"#F'*$%\"aGF,F',&%\"nGF'F'!\"\"F1F+" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^n),x) = x/((2*n-2)*a^2*(x^2+a^2)^ (n-1))+(2*n-3)/((2*n-2)*a^2);" "6#/-%$IntG6$*&\"\"\"F(),&*$%\"xG\"\"#F (*$%\"aGF-F(%\"nG!\"\"F,,&*&F,F(*(,&*&F-F(F0F(F(F-F1F(*$F/F-F(),&*$F,F -F(*$F/F-F(,&F0F(F(F1F(F1F(*&,&*&F-F(F0F(F(\"\"$F1F(*&,&*&F-F(F0F(F(F- F1F(*$F/F-F(F1F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^(n- 1)),x)" "6#-%$IntG6$*&\"\"\"F'),&*$%\"xG\"\"#F'*$%\"aGF,F',&%\"nGF'F'! \"\"F1F+" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 322 48 "________________________________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "n=2" "6#/%\"nG\"\"# " }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/((x^2+a^2)^2),x) = x/(2*a^2*(x^2+a^2))+1/(2*a^2); " "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF-F(F-!\"\"F,,&*&F,F (*(F-F(*$F/F-F(,&*$F,F-F(*$F/F-F(F(F0F(*&F(F(*&F-F(*$F/F-F(F0F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)),x)" "6#-%$IntG6$*&\" \"\"F',&*$%\"xG\"\"#F'*$%\"aGF+F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/(2*a^2*(x^2+a^2))+1/(2*a ^3);" "6#/%!G,&*&%\"xG\"\"\"*(\"\"#F(*$%\"aGF*F(,&*$F'F*F(*$F,F*F(F(! \"\"F(*&F(F(*&F*F(*$F,\"\"$F(F0F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arc tan(x/a) + c" "6#,&-%'arctanG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "n = 3;" "6#/%\"nG\"\"$" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^3),x) = x/(4*a^2*(x^2 +a^2)^2)+3/(4*a^2);" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF -F(\"\"$!\"\"F,,&*&F,F(*(\"\"%F(*$F/F-F(,&*$F,F-F(*$F/F-F(F-F1F(*&F0F( *&F5F(*$F/F-F(F1F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^2 ),x);" "6#-%$IntG6$*&\"\"\"F'*$,&*$%\"xG\"\"#F'*$%\"aGF,F'F,!\"\"F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/(4*a^2*(x^2+a^2)^2)+3/(4*a^2);" "6#/%!G,&*&%\"xG\"\"\"*(\"\"%F(*$%\"aG\"\"#F(,&*$F'F-F(*$F,F-F(F-!\"\" F(*&\"\"$F(*&F*F(*$F,F-F(F1F(" }{XPPEDIT 18 0 "``(x/(2*a^2*(x^2+a^2))+ ``(1/(2*a^3))*arctan(x/a))+c;" "6#,&-%!G6#,&*&%\"xG\"\"\"*(\"\"#F**$% \"aGF,F*,&*$F)F,F**$F.F,F*F*!\"\"F**&-F%6#*&F*F**&F,F**$F.\"\"$F*F2F*- %'arctanG6#*&F)F*F.F2F*F*F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ x/(4*a^2*(x^2+a^2)^2)+3*x/(8*a^4*(x^2+a^2))+3/(8*a^5);" "6#/%!G,(*&%\" xG\"\"\"*(\"\"%F(*$%\"aG\"\"#F(,&*$F'F-F(*$F,F-F(F-!\"\"F(*(\"\"$F(F'F (*(\"\")F(*$F,F*F(,&*$F'F-F(*$F,F-F(F(F1F(*&F3F(*&F5F(*$F,\"\"&F(F1F( " }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c" "6#,&-%'arctanG6#*&% \"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "n=4" "6#/%\"nG\"\"%" }{TEXT -1 9 " \+ we have " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(1/(( x^2+a^2)^4),x) = x/(6*a^2*(x^2+a^2)^3)+5/(6*a^2);" "6#/-%$IntG6$*&\"\" \"F(*$,&*$%\"xG\"\"#F(*$%\"aGF-F(\"\"%!\"\"F,,&*&F,F(*(\"\"'F(*$F/F-F( ,&*$F,F-F(*$F/F-F(\"\"$F1F(*&\"\"&F(*&F5F(*$F/F-F(F1F(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^3),x);" "6#-%$IntG6$*&\"\"\"F'*$,&* $%\"xG\"\"#F'*$%\"aGF,F'\"\"$!\"\"F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= x/(6*a^2*(x^2+a^2)^3)+5/(6*a^2)" "6#/%!G,&*&%\"xG\"\"\"*(\"\"'F(*$%\"a G\"\"#F(,&*$F'F-F(*$F,F-F(\"\"$!\"\"F(*&\"\"&F(*&F*F(*$F,F-F(F2F(" } {XPPEDIT 18 0 " ``(x/(4*a^2*(x^2+a^2)^2)+3*x/(8*a^4*(x^2+a^2))+``(3/(8 *a^5))*arctan(x/a))+c" "6#,&-%!G6#,(*&%\"xG\"\"\"*(\"\"%F**$%\"aG\"\"# F*,&*$F)F/F**$F.F/F*F/!\"\"F**(\"\"$F*F)F**(\"\")F**$F.F,F*,&*$F)F/F** $F.F/F*F*F3F**&-F%6#*&F5F**&F7F**$F.\"\"&F*F3F*-%'arctanG6#*&F)F*F.F3F *F*F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x/(6*a^2*(x^2+a^2)^3)+ 5*x/(24*a^4*(x^2+a^2)^2)+5*x/(16*a^6*(x^2+a^2))+5/(16*a^7);" "6#/%!G,* *&%\"xG\"\"\"*(\"\"'F(*$%\"aG\"\"#F(,&*$F'F-F(*$F,F-F(\"\"$!\"\"F(*(\" \"&F(F'F(*(\"#CF(*$F,\"\"%F(,&*$F'F-F(*$F,F-F(F-F2F(*(F4F(F'F(*(\"#;F( *$F,F*F(,&*$F'F-F(*$F,F-F(F(F2F(*&F4F(*&F>F(*$F,\"\"(F(F2F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a)+c" "6#,&-%'arctanG6#*&%\"xG\"\"\" %\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Int(1/(x^2+a^2)^4,x);\nv alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$),&*$ )%\"aG\"\"#F'F'*$)%\"xGF.F'F'\"\"%F'!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***\"\"'!\"\"%\"xG\"\"\"%\"aG!\"#,&*$)F)\"\"#F(F(*$)F' F.F(F(!\"$F(*,\"\"&F(\"#CF&F)!\"%F'F(F+F*F(*,F3F(\"#;F&F)!\"'F'F(F+F&F (*&#F3F7F(*&F)!\"(-%'arctanG6#*&F'F(F)F&F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 100 "Partial fraction expansions \+ which involve repeated irreducible quadratic factors in the denominato r " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 32 "A rational function of the form " }{XPPEDIT 18 0 "P(x)/(( a*x^2+b*x+c)^n);" "6#*&-%\"PG6#%\"xG\"\"\"),(*&%\"aGF(*$F'\"\"#F(F(*&% \"bGF(F'F(F(%\"cGF(%\"nG!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 " a*x^2+b*x+c" "6#,(*&%\"aG\"\"\"*$%\"xG\"\"#F&F&*&%\"bGF&F(F&F&%\"cGF& " }{TEXT -1 7 " is an " }{TEXT 259 21 "irreducible quadratic" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P(x)" "6#-%\"PG6#%\"xG" }{TEXT -1 37 " is a polynomial of degree less than " }{XPPEDIT 18 0 "2*n" "6#*&\"\"#\" \"\"%\"nGF%" }{TEXT -1 20 ", can be expanded as" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P(x)/((a*x^2+b*x+c)^n) = (A[1]*x+B[1])/ (a*x^2+b*x+c)+(A[2]*x+B[2])/((a*x^2+b*x+c)^2)+` . . . `+(A[n]*x+B[n])/ ((a*x^2+b*x+c)^n);" "6#/*&-%\"PG6#%\"xG\"\"\"),(*&%\"aGF)*$F(\"\"#F)F) *&%\"bGF)F(F)F)%\"cGF)%\"nG!\"\",**&,&*&&%\"AG6#F)F)F(F)F)&%\"BG6#F)F) F),(*&F-F)*$F(F/F)F)*&F1F)F(F)F)F2F)F4F)*&,&*&&F:6#F/F)F(F)F)&F=6#F/F) F)*$,(*&F-F)*$F(F/F)F)*&F1F)F(F)F)F2F)F/F4F)%(~.~.~.~GF)*&,&*&&F:6#F3F )F(F)F)&F=6#F3F)F)),(*&F-F)*$F(F/F)F)*&F1F)F(F)F)F2F)F3F4F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{PARA 0 "" 0 "" {TEXT 264 8 "Question" }{TEXT -1 11 ": Find " } {XPPEDIT 18 0 "Int((x^3+1)/((x^2+2*x+2)^2),x);" "6#-%$IntG6$*&,&*$%\"x G\"\"$\"\"\"F+F+F+*$,(*$F)\"\"#F+*&F/F+F)F+F+F/F+F/!\"\"F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The integrand has a partial fraction expansion of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^3+1 )/((x^2+2*x+2)^2) = (A*x+B)/(x^2+2*x+2)+(C*x+`D `)/((x^2+2*x+2)^2);" " 6#/*&,&*$%\"xG\"\"$\"\"\"F)F)F)*$,(*$F'\"\"#F)*&F-F)F'F)F)F-F)F-!\"\", &*&,&*&%\"AGF)F'F)F)%\"BGF)F),(*$F'F-F)*&F-F)F'F)F)F-F)F/F)*&,&*&%\"CG F)F'F)F)%#D~GF)F)*$,(*$F'F-F)*&F-F)F'F)F)F-F)F-F/F)" }{TEXT -1 15 " - ------ (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Multiplying both sides of (i) by " }{XPPEDIT 18 0 "(x^2+2*x+2)^ 2" "6#*$,(*$%\"xG\"\"#\"\"\"*&F'F(F&F(F(F'F(F'" }{TEXT -1 8 " gives: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3+1 = (A*x+B)*( x^2+2*x+2)+C*x+`D `;" "6#/,&*$%\"xG\"\"$\"\"\"F(F(,(*&,&*&%\"AGF(F&F(F (%\"BGF(F(,(*$F&\"\"#F(*&F1F(F&F(F(F1F(F(F(*&%\"CGF(F&F(F(%#D~GF(" } {TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " }{TEXT 294 1 "A" }{TEXT -1 2 ", " }{TEXT 295 1 "B" }{TEXT -1 2 ", " }{TEXT 296 1 "C" }{TEXT -1 5 " and " }{TEXT 297 1 "D" }{TEXT -1 49 " by the two str ategies of substituting values of " }{TEXT 292 1 "x" }{TEXT -1 48 " in (ii) and equating coefficients of powers of " }{TEXT 293 1 "x" } {TEXT -1 30 " on the left and right sides. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^3]*` . . . `;" "6#*& 7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"$F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "1 = A;" "6#/\"\"\"%\"AG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^2] *` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"#F'%(~.~.~. ~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*A+B;" "6#/\"\"!,&*&\"\"# \"\"\"%\"AGF(F(%\"BGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x]*` . . . `;" "6#*&7#*(%-coeff icientsG\"\"\"%#ofGF'%\"xGF'F'%(~.~.~.~GF'" }{TEXT -1 3 " " } {XPPEDIT 18 0 "0 = 2*A+2*B+C;" "6#/\"\"!,(*&\"\"#\"\"\"%\"AGF(F(*&F'F( %\"BGF(F(%\"CGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[x = 0]*` . . . `;" "6#*&7#/%\"xG\"\"!\"\"\"%(~.~.~.~GF (" }{TEXT -1 5 " " }{XPPEDIT 18 0 "1 = 2*B+D;" "6#/\"\"\",&*&\"\"# F$%\"BGF$F$%\"DGF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Sub stituting " }{XPPEDIT 18 0 "A=1" "6#/%\"AG\"\"\"" }{TEXT -1 30 " in th e second equation gives " }{XPPEDIT 18 0 "B=-2" "6#/%\"BG,$\"\"#!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " } {XPPEDIT 18 0 "A=1" "6#/%\"AG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B=-2" "6#/%\"BG,$\"\"#!\"\"" }{TEXT -1 29 " in the third equation g ives " }{XPPEDIT 18 0 "0 = 2-4+C;" "6#/\"\"!,(\"\"#\"\"\"\"\"%!\"\"%\" CGF'" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "C=2" "6#/%\"CG\"\"#" } {TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " } {XPPEDIT 18 0 "B=-2" "6#/%\"BG,$\"\"#!\"\"" }{TEXT -1 28 " in the last equation gives " }{XPPEDIT 18 0 "1=-4+`D `" "6#/\"\"\",&\"\"%!\"\"%#D ~GF$" }{TEXT -1 7 "so that" }{XPPEDIT 18 0 "` D`=5" "6#/%#~DG\"\"&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Thus we obtain the partial fraction expansion " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^3+1)/((x^2+2*x+2)^2) = ( x-2)/(x^2+2*x+2)+(2*x+5)/((x^2+2*x+2)^2)" "6#/*&,&*$%\"xG\"\"$\"\"\"F) F)F)*$,(*$F'\"\"#F)*&F-F)F'F)F)F-F)F-!\"\",&*&,&F'F)F-F/F),(*$F'F-F)*& F-F)F'F)F)F-F)F/F)*&,&*&F-F)F'F)F)\"\"&F)F)*$,(*$F'F-F)*&F-F)F'F)F)F-F )F-F/F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means th at " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x^3+1)/(( x^2+2*x+2)^2),x)=Int((x-2)/(x^2+2*x+2),x)+Int((2*x+5)/((x^2+2*x+2)^2), x)" "6#/-%$IntG6$*&,&*$%\"xG\"\"$\"\"\"F,F,F,*$,(*$F*\"\"#F,*&F0F,F*F, F,F0F,F0!\"\"F*,&-F%6$*&,&F*F,F0F2F,,(*$F*F0F,*&F0F,F*F,F,F0F,F2F*F,-F %6$*&,&*&F0F,F*F,F,\"\"&F,F,*$,(*$F*F0F,*&F0F,F*F,F,F0F,F0F2F*F," } {TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "We determine the two integrals on the rig ht separately. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t((x-2)/(x^2+2*x+2),x) = Int((x-2)/((x+1)^2+1),x)" "6#/-%$IntG6$*&,&% \"xG\"\"\"\"\"#!\"\"F*,(*$F)F+F**&F+F*F)F*F*F+F*F,F)-F%6$*&,&F)F*F+F,F *,&*$,&F)F*F*F*F+F*F*F*F,F)" }{TEXT -1 6 " --- " }{XPPEDIT 18 0 "PIEC EWISE([u=x+1,x=u-1],[``,``],[du=dx,``])" "6#-%*PIECEWISEG6%7$/%\"uG,&% \"xG\"\"\"F+F+/F*,&F(F+F+!\"\"7$%!GF07$/%#duG%#dxGF0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int((u-3)/(u^2+1 ),u)" "6#/%!G-%$IntG6$*&,&%\"uG\"\"\"\"\"$!\"\"F+,&*$F*\"\"#F+F+F+F-F* " }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` `=Int(u/(u^2+1),u)-3*Int(1/(u^2+1),u)" "6#/%!G,&-%$IntG6$*&%\"uG\"\"\" ,&*$F*\"\"#F+F+F+!\"\"F*F+*&\"\"$F+-F'6$*&F+F+,&*$F*F.F+F+F+F/F*F+F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(u^2+1)-3*arctan(u)+c[1]" "6#,(-%# lnG6#,&*$%\"uG\"\"#\"\"\"F+F+F+*&\"\"$F+-%'arctanG6#F)F+!\"\"&%\"cG6#F +F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-2)/(x^2+2*x+2),x)=1/ 2" "6#/-%$IntG6$*&,&%\"xG\"\"\"\"\"#!\"\"F*,(*$F)F+F**&F+F*F)F*F*F+F*F ,F)*&F*F*F+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+2*x+2)-3*arctan( x+1)+c[1]" "6#,(-%#lnG6#,(*$%\"xG\"\"#\"\"\"*&F*F+F)F+F+F*F+F+*&\"\"$F +-%'arctanG6#,&F)F+F+F+F+!\"\"&%\"cG6#F+F+" }{TEXT -1 16 " ------- (i v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "A lternatively, noting that " }{XPPEDIT 18 0 "Diff([x^2+2*x+2],x) = 2*x +2;" "6#/-%%DiffG6$7#,(*$%\"xG\"\"#\"\"\"*&F+F,F*F,F,F+F,F*,&*&F+F,F*F ,F,F+F," }{TEXT -1 44 " we try to expand the integrand in the form " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-2)/(x^2+2*x+2)=P* (2*x+2)/(x^2+2*x+2)+Q/(x^2+2*x+2)" "6#/*&,&%\"xG\"\"\"\"\"#!\"\"F',(*$ F&F(F'*&F(F'F&F'F'F(F'F),&*(%\"PGF',&*&F(F'F&F'F'F(F'F',(*$F&F(F'*&F(F 'F&F'F'F(F'F)F'*&%\"QGF',(*$F&F(F'*&F(F'F&F'F'F(F'F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "P and Q must have values so that " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-2=P*(2*x+2)+Q" "6# /,&%\"xG\"\"\"\"\"#!\"\",&*&%\"PGF&,&*&F'F&F%F&F&F'F&F&F&%\"QGF&" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "identically. " }}{PARA 0 "" 0 "" {TEXT -1 30 "Comparing the coefficients of " }{TEXT 273 1 "x " }{TEXT -1 57 " on the left and right sides of this equation shows th at " }{XPPEDIT 18 0 "2*P=1" "6#/*&\"\"#\"\"\"%\"PGF&F&" }{TEXT -1 9 " \+ so that " }{XPPEDIT 18 0 "P=1/2" "6#/%\"PG*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 " x=0" "6#/%\"xG\"\"!" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "-2=2*P+Q" " 6#/,$\"\"#!\"\",&*&F%\"\"\"%\"PGF)F)%\"QGF)" }{TEXT -1 9 " so that " } {XPPEDIT 18 0 "Q=-3" "6#/%\"QG,$\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "u nassign('x','P','Q');\neq := (x-2)/(x^2+2*x+2)=P*(2*x+2)/(x^2+2*x+2)+Q /(x^2+2*x+2);\nsolve(identity(eq,x),\{P,Q\});\nassign(%);\neq;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/*&,&%\"xG\"\"\"\"\"#!\"\"F),(*$ )F(F*F)F)*&F*F)F(F)F)F*F)F+,&*(%\"PGF),&*&F*F)F(F)F)F*F)F)F,F+F)*&%\"Q GF)F,F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"PG#\"\"\"\"\"#/%\"Q G!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&%\"xG\"\"\"\"\"#!\"\"F', (*$)F&F(F'F'*&F(F'F&F'F'F(F'F),&*(F(F),&*&F(F'F&F'F'F(F'F'F*F)F'*&\"\" $F'F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-2)/(x^2+2*x+2) ,x)=1/2" "6#/-%$IntG6$*&,&%\"xG\"\"\"\"\"#!\"\"F*,(*$F)F+F**&F+F*F)F*F *F+F*F,F)*&F*F*F+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+2)/(x^2+ 2*x+2),x)-3*Int(1/(x^2+2*x+2),x)" "6#,&-%$IntG6$*&,&*&\"\"#\"\"\"%\"xG F+F+F*F+F+,(*$F,F*F+*&F*F+F,F+F+F*F+!\"\"F,F+*&\"\"$F+-F%6$*&F+F+,(*$F ,F*F+*&F*F+F,F+F+F*F+F0F,F+F0" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+2*x+2)-3*arctan(x+1)+c[1]" "6#,( -%#lnG6#,(*$%\"xG\"\"#\"\"\"*&F*F+F)F+F+F*F+F+*&\"\"$F+-%'arctanG6#,&F )F+F+F+F+!\"\"&%\"cG6#F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "as before." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The second integral " }{XPPEDIT 18 0 "Int((2*x+5)/((x^2+2 *x+2)^2),x)" "6#-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF*F*\"\"&F*F**$,(*$F+F) F**&F)F*F+F*F*F)F*F)!\"\"F+" }{TEXT -1 68 " in (iii) can be tackled al ong similar lines to the first integral. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+5)/(x ^2+2*x+2),x) = Int((2*x+5)/(((x+1)^2+1)^2),x);" "6#/-%$IntG6$*&,&*&\" \"#\"\"\"%\"xGF+F+\"\"&F+F+,(*$F,F*F+*&F*F+F,F+F+F*F+!\"\"F,-F%6$*&,&* &F*F+F,F+F+F-F+F+*$,&*$,&F,F+F+F+F*F+F+F+F*F1F," }{TEXT -1 6 " --- " }{XPPEDIT 18 0 "PIECEWISE([u=x+1,x=u-1],[``,``],[du=dx,``])" "6#-%*PIE CEWISEG6%7$/%\"uG,&%\"xG\"\"\"F+F+/F*,&F(F+F+!\"\"7$%!GF07$/%#duG%#dxG F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((2*u+3)/((u^2+1)^2),u);" "6# /%!G-%$IntG6$*&,&*&\"\"#\"\"\"%\"uGF,F,\"\"$F,F,*$,&*$F-F+F,F,F,F+!\" \"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=Int((2*u)/((u^2+1)^2),u)" "6#/% !G-%$IntG6$*(\"\"#\"\"\"%\"uGF**$,&*$F+F)F*F*F*F)!\"\"F+" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "3*Int(1/((u^2+1)^2),u)" "6#*&\"\"$\"\"\"-%$IntG6 $*&F%F%*$,&*$%\"uG\"\"#F%F%F%F.!\"\"F-F%" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " The first of thes e two integrals of expressions in " }{TEXT 266 1 "u" }{TEXT -1 43 " ca n be found by means of the substitution " }{XPPEDIT 18 0 "v=u^2+1" "6# /%\"vG,&*$%\"uG\"\"#\"\"\"F)F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(2*u/((u^2+1)^2),u)" "6#-%$IntG6$*( \"\"#\"\"\"%\"uGF(*$,&*$F)F'F(F(F(F'!\"\"F)" }{TEXT -1 7 " --- " } {XPPEDIT 18 0 "PIECEWISE([v = u^2+1, ``],[``, ``],[dv = 2*u*du, ``]); " "6#-%*PIECEWISEG6%7$/%\"vG,&*$%\"uG\"\"#\"\"\"F-F-%!G7$F.F.7$/%#dvG* (F,F-F+F-%#duGF-F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=Int(1/v^2,v)" "6#/%!G-%$IntG6$*&\"\"\"F)*$%\"vG\" \"#!\"\"F+" }{XPPEDIT 18 0 " ``=-1/v+c[3]" "6#/%!G,&*&\"\"\"F'%\"vG!\" \"F)&%\"cG6#\"\"$F'" }{XPPEDIT 18 0 " ``=-1/(u^2+1)+c[3]" "6#/%!G,&*& \"\"\"F',&*$%\"uG\"\"#F'F'F'!\"\"F,&%\"cG6#\"\"$F'" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 30 "On the other hand the formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^2),x)=x/(2 *a^2*(a^2+x^2))+1/(2*a^3)" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$ %\"aGF-F(F-!\"\"F,,&*&F,F(*(F-F(*$F/F-F(,&*$F/F-F(*$F,F-F(F(F0F(*&F(F( *&F-F(*$F/\"\"$F(F0F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a) +c " "6#,&-%'arctanG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "derived earlier in this section gives \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((u^2+1)^2) ,u) = u/(2*(u^2+1))+1/2;" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"uG\"\"#F(F(F (F-!\"\"F,,&*&F,F(*&F-F(,&*$F,F-F(F(F(F(F.F(*&F(F(F-F.F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(u) + c[4]" "6#,&-%'arctanG6#%\"uG\"\"\"&%\" cG6#\"\"%F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+5)/((x^2+2*x +2)^2),x) = -1/(u^2+1)+3*u/(2*(u^2+1))+3/2;" "6#/-%$IntG6$*&,&*&\"\"# \"\"\"%\"xGF+F+\"\"&F+F+*$,(*$F,F*F+*&F*F+F,F+F+F*F+F*!\"\"F,,(*&F+F+, &*$%\"uGF*F+F+F+F2F2*(\"\"$F+F7F+*&F*F+,&*$F7F*F+F+F+F+F2F+*&F9F+F*F2F +" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(u) + c[5]" "6#,&-%'arctanG6# %\"uG\"\"\"&%\"cG6#\"\"&F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/(x^ 2+2*x+2)+(3*x+3)/(2*(x^2+2*x+2))+3/2;" "6#/%!G,(*&\"\"\"F',(*$%\"xG\" \"#F'*&F+F'F*F'F'F+F'!\"\"F-*&,&*&\"\"$F'F*F'F'F1F'F'*&F+F',(*$F*F+F'* &F+F'F*F'F'F+F'F'F-F'*&F1F'F+F-F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arc tan(x+1) + c[5]" "6#,&-%'arctanG6#,&%\"xG\"\"\"F)F)F)&%\"cG6#\"\"&F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+5)/((x^2+2*x+2)^2),x)=(3*x+1)/(2*(x^2+2*x+2))+3/2" "6#/-%$ IntG6$*&,&*&\"\"#\"\"\"%\"xGF+F+\"\"&F+F+*$,(*$F,F*F+*&F*F+F,F+F+F*F+F *!\"\"F,,&*&,&*&\"\"$F+F,F+F+F+F+F+*&F*F+,(*$F,F*F+*&F*F+F,F+F+F*F+F+F 2F+*&F7F+F*F2F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x+1)+c[5]" "6# ,&-%'arctanG6#,&%\"xG\"\"\"F)F)F)&%\"cG6#\"\"&F)" }{TEXT -1 15 " ---- --- (v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The 2nd integral " }{XPPEDIT 18 0 "Int((2*x+5)/((x^2+2*x+2)^2),x) " "6#-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF*F*\"\"&F*F**$,(*$F+F)F**&F)F*F+F *F*F)F*F)!\"\"F+" }{TEXT -1 147 " on the right side of (iii) can also \+ be tackled by an alternative method along similar lines to the alterna tive method used for the first integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([x/(x^2+2*x+2)],x) = (1*`.`*( x^2+2*x+2)-x*`.`*(2*x+2))/((x^2+2*x+2)^2);" "6#/-%%DiffG6$7#*&%\"xG\" \"\",(*$F)\"\"#F**&F-F*F)F*F*F-F*!\"\"F)*&,&*(F*F*%\".GF*,(*$F)F-F**&F -F*F)F*F*F-F*F*F**(F)F*F3F*,&*&F-F*F)F*F*F-F*F*F/F**$,(*$F)F-F**&F-F*F )F*F*F-F*F-F/" }{XPPEDIT 18 0 "``=(2-x^2)/((x^2+2*x+2)^2)" "6#/%!G*&,& \"\"#\"\"\"*$%\"xGF'!\"\"F(*$,(*$F*F'F(*&F'F(F*F(F(F'F(F'F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([1/(x^2+2*x+2)],x) = -(2*x+2)/((x^ 2+2*x+2)^2);" "6#/-%%DiffG6$7#*&\"\"\"F),(*$%\"xG\"\"#F)*&F-F)F,F)F)F- F)!\"\"F,,$*&,&*&F-F)F,F)F)F-F)F)*$,(*$F,F-F)*&F-F)F,F)F)F-F)F-F/F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "We now attempt to find \+ an expansion in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(2*x+5)/((x^2+2*x+2)^2)=P*(2-x^2)/((x^2+2*x+2)^2)-Q*(2* x+2)/((x^2+2*x+2)^2)+R/(x^2+2*x+2)" "6#/*&,&*&\"\"#\"\"\"%\"xGF(F(\"\" &F(F(*$,(*$F)F'F(*&F'F(F)F(F(F'F(F'!\"\",(*(%\"PGF(,&F'F(*$F)F'F/F(*$, (*$F)F'F(*&F'F(F)F(F(F'F(F'F/F(*(%\"QGF(,&*&F'F(F)F(F(F'F(F(*$,(*$F)F' F(*&F'F(F)F(F(F'F(F'F/F/*&%\"RGF(,(*$F)F'F(*&F'F(F)F(F(F'F(F/F(" } {TEXT -1 16 " ------- (vi). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "Multiplying equation (vi) by " }{XPPEDIT 18 0 "(x^2+2*x+2)^2" "6#*$,(*$%\"xG\"\"#\"\"\"*&F'F(F&F(F(F'F(F'" } {TEXT -1 7 "gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*x+5=P*(2-x^2)-Q*(2*x+2)+R*(x^2+2*x+2)" "6#/,&*&\"\"#\"\"\"%\"xGF' F'\"\"&F',(*&%\"PGF',&F&F'*$F(F&!\"\"F'F'*&%\"QGF',&*&F&F'F(F'F'F&F'F' F/*&%\"RGF',(*$F(F&F'*&F&F'F(F'F'F&F'F'F'" }{TEXT -1 17 " ------- (vi i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "W e can obtain equations involving " }{TEXT 268 1 "P" }{TEXT -1 2 ", " } {TEXT 269 1 "Q" }{TEXT -1 5 " and " }{TEXT 270 1 "R" }{TEXT -1 27 " by substituting values of " }{TEXT 271 1 "x" }{TEXT -1 49 " in (vii) and equating coefficients of powers of " }{TEXT 272 1 "x" }{TEXT -1 30 " \+ on the left and right sides. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[coefficients*of*x^2]*` . . . `;" "6#*&7#*(%-coefficien tsG\"\"\"%#ofGF'%\"xG\"\"#F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = -P+R;" "6#/\"\"!,&%\"PG!\"\"%\"RG\"\"\"" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xGF'F'%(~.~.~.~GF' " }{TEXT -1 5 " " }{XPPEDIT 18 0 "2 = -2*Q+2*R;" "6#/\"\"#,&*&F$\" \"\"%\"QGF'!\"\"*&F$F'%\"RGF'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[x = 0]*` . . . `;" "6#*&7#/%\"xG\"\"! \"\"\"%(~.~.~.~GF(" }{TEXT -1 3 " " }{XPPEDIT 18 0 "5 = 2*P-2*Q+2*R; " "6#/\"\"&,(*&\"\"#\"\"\"%\"PGF(F(*&F'F(%\"QGF(!\"\"*&F'F(%\"RGF(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "The first equation give s " }{XPPEDIT 18 0 "P=R" "6#/%\"PG%\"RG" }{TEXT -1 59 " and dividing b oth sides of the second equation by 2 gives " }{XPPEDIT 18 0 "R-Q=1" " 6#/,&%\"RG\"\"\"%\"QG!\"\"F&" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 " R=Q+1" "6#/%\"RG,&%\"QG\"\"\"F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "The third equation now implies that " }{XPPEDIT 18 0 "4*R -2*Q=5" "6#/,&*&\"\"%\"\"\"%\"RGF'F'*&\"\"#F'%\"QGF'!\"\"\"\"&" } {TEXT -1 9 " so that " }{XPPEDIT 18 0 "4*Q+4-2*Q=5" "6#/,(*&\"\"%\"\" \"%\"QGF'F'F&F'*&\"\"#F'F(F'!\"\"\"\"&" }{TEXT -1 8 " giving " } {XPPEDIT 18 0 "2*Q=1" "6#/*&\"\"#\"\"\"%\"QGF&F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Q=1/2" "6#/%\"QG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 ". T hen " }{XPPEDIT 18 0 "R=3/2" "6#/%\"RG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P=3/2" "6#/%\"PG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "Substituting the value s for " }{TEXT 274 1 "P" }{TEXT -1 2 ", " }{TEXT 275 1 "Q" }{TEXT -1 5 " and " }{TEXT 276 1 "R" }{TEXT -1 16 " in (vii) gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(2*x+5)/((x^2+2*x+2)^2) = 3 *(2-x^2)/(2*(x^2+2*x+2)^2)-(2*x+2)/(2*(x^2+2*x+2)^2)+3/(2*(x^2+2*x+2)) ;" "6#/*&,&*&\"\"#\"\"\"%\"xGF(F(\"\"&F(F(*$,(*$F)F'F(*&F'F(F)F(F(F'F( F'!\"\",(*(\"\"$F(,&F'F(*$F)F'F/F(*&F'F(*$,(*$F)F'F(*&F'F(F)F(F(F'F(F' F(F/F(*&,&*&F'F(F)F(F(F'F(F(*&F'F(*$,(*$F)F'F(*&F'F(F)F(F(F'F(F'F(F/F/ *&F2F(*&F'F(,(*$F)F'F(*&F'F(F)F(F(F'F(F(F/F(" }{TEXT -1 16 " ------- \+ (vi). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "unassign('x','P','Q','R');\neq := (2*x+5)/(x^2+2*x+2 )^2=P*(2-x^2)/(x^2+2*x+2)^2-Q*(2*x+2)/(x^2+2*x+2)^2+R/(x^2+2*x+2);\nso lve(identity(eq,x),\{P,Q,R\});\nassign(%);\neq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/*&,&*&\"\"#\"\"\"%\"xGF*F*\"\"&F*F*,(*$)F+F)F*F* *&F)F*F+F*F*F)F*!\"#,(*(%\"PGF*,&F)F*F.!\"\"F*F-F1F**(%\"QGF*,&*&F)F*F +F*F*F)F*F*F-F1F6*&%\"RGF*F-F6F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<% /%\"QG#\"\"\"\"\"#/%\"PG#\"\"$F(/%\"RGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*&\"\"#\"\"\"%\"xGF(F(\"\"&F(F(,(*$)F)F'F(F(*&F'F(F)F(F(F'F (!\"#,(**\"\"$F(F'!\"\",&F'F(F,F3F(F+F/F(*(F'F3,&*&F'F(F)F(F(F'F(F(F+F /F3*(F2F(F'F3F+F3F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We now h ave " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+5)/( (x^2+2*x+2)^2),x) = 3/2" "6#/-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF+F+\"\"&F +F+*$,(*$F,F*F+*&F*F+F,F+F+F*F+F*!\"\"F,*&\"\"$F+F*F2" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int((2-x^2)/((x^2+2*x+2)^2),x)+1/2;" "6#,&-%$IntG6$* &,&\"\"#\"\"\"*$%\"xGF)!\"\"F**$,(*$F,F)F**&F)F*F,F*F*F)F*F)F-F,F**&F* F*F)F-F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-(2*x+2))/((x^2+2*x+2)^ 2),x)+3/2;" "6#,&-%$IntG6$*&,$,&*&\"\"#\"\"\"%\"xGF,F,F+F,!\"\"F,*$,(* $F-F+F,*&F+F,F-F,F,F+F,F+F.F-F,*&\"\"$F,F+F.F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/(x^2+2*x+2),x)" "6#-%$IntG6$*&\"\"\"F',(*$%\"xG\" \"#F'*&F+F'F*F'F'F+F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3*x/(2 *(x^2+2*x+2))+1/2(x^2+2*x+2)+3/2;" "6#/%!G,(*(\"\"$\"\"\"%\"xGF(*&\"\" #F(,(*$F)F+F(*&F+F(F)F(F(F+F(F(!\"\"F(*&F(F(-F+6#,(*$F)F+F(*&F+F(F)F(F (F+F(F/F(*&F'F(F+F/F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x+1)+c[5 ];" "6#,&-%'arctanG6#,&%\"xG\"\"\"F)F)F)&%\"cG6#\"\"&F)" }{TEXT -1 2 " , " }}{PARA 257 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((2*x+5)/((x^2+2*x+2)^2),x)=(3*x+1)/ (2*(x^2+2*x+2))+3/2" "6#/-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF+F+\"\"&F+F+* $,(*$F,F*F+*&F*F+F,F+F+F*F+F*!\"\"F,,&*&,&*&\"\"$F+F,F+F+F+F+F+*&F*F+, (*$F,F*F+*&F*F+F,F+F+F*F+F+F2F+*&F7F+F*F2F+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(x+1)+c[5]" "6#,&-%'arctanG6#,&%\"xG\"\"\"F)F)F)& %\"cG6#\"\"&F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "which is the result (v) obtained previous ly" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Com bining the results from (iv) and (v) gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int((x^3+1)/((x^2+2*x+2)^2),x) = Int(( x-2)/(x^2+2*x+2),x) + Int((2*x+5)/((x^2+2*x+2)^2),x)" "6#/-%$IntG6$*&, &*$%\"xG\"\"$\"\"\"F,F,F,*$,(*$F*\"\"#F,*&F0F,F*F,F,F0F,F0!\"\"F*,&-F% 6$*&,&F*F,F0F2F,,(*$F*F0F,*&F0F,F*F,F,F0F,F2F*F,-F%6$*&,&*&F0F,F*F,F, \"\"&F,F,*$,(*$F*F0F,*&F0F,F*F,F,F0F,F0F2F*F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "ln(x^2+2*x+2)-3*arctan(x+1) +(3*x+1)/(2*(x^2+2*x+2)) +3/2" "6#,*-%#lnG6#,(*$%\"xG\"\"#\"\"\"*&F*F+F)F+F+F*F+F+*&\"\"$F+-%'a rctanG6#,&F)F+F+F+F+!\"\"*&,&*&F.F+F)F+F+F+F+F+*&F*F+,(*$F)F*F+*&F*F+F )F+F+F*F+F+F3F+*&F.F+F*F3F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x+ 1) + c" "6#,&-%'arctanG6#,&%\"xG\"\"\"F)F)F)%\"cGF)" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "ln(x^2+2*x+2)+(3*x+1)/(2*(x^2+2*x+2))-3/2" "6#,(-%#l nG6#,(*$%\"xG\"\"#\"\"\"*&F*F+F)F+F+F*F+F+*&,&*&\"\"$F+F)F+F+F+F+F+*&F *F+,(*$F)F*F+*&F*F+F)F+F+F*F+F+!\"\"F+*&F0F+F*F5F5" }{TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(x+1) + c" "6#,&-%'arctanG6#,&%\"xG\"\"\"F)F)F)% \"cGF)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 267 33 "_________________________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "Some of the step s used to find this integral are given by the following Maple commands . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "Int((x^3+1)/((x^2+2*x+2)^2),x);\n``=map(convert,%,pa rfrac,x);\n``=map(Int,op(1,rhs(%)),x);\n``=map(``,rhs(%)):\n``=value(r hs(%));\n``=map(simplify,eval(subs(``=(_u->_u),rhs(%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*$)%\"xG\"\"$\"\"\"F,F,F,F,,(*$)F* \"\"#F,F,*&F0F,F*F,F,F0F,!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G-%$IntG6$,&*&,&\"\"&\"\"\"*&\"\"#F,%\"xGF,F,F,,(*$)F/F.F,F,*&F.F,F/F, F,F.F,!\"#F,*&,&F/F,F.!\"\"F,F0F7F,F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$IntG6$*&,&\"\"&\"\"\"*&\"\"#F,%\"xGF,F,F,,(*$)F/F.F,F,*&F. F,F/F,F,F.F,!\"#F/F,-F'6$*&,&F/F,F.!\"\"F,F0F9F/F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,&-F$6#,&*(\"\"%!\"\",&*&\"\"'\"\"\"%\"xGF/F/\"\" #F/F/,(*$)F0F1F/F/*&F1F/F0F/F/F1F/F+F/*&#\"\"$F1F/-%'arctanG6#,&F0F/F/ F/F/F/F/-F$6#,&*&#F/F1F/-%#lnG6#F2F/F/*&F8F/F9F/F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"#!\"\",&*&\"\"$\"\"\"%\"xGF,F,F,F,F,,(*$ )F-F'F,F,*&F'F,F-F,F,F'F,F(F,*&#F+F'F,-%'arctanG6#,&F-F,F,F,F,F(*&#F,F 'F,-%#lnG6#F.F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ex ample 2" }}{PARA 0 "" 0 "" {TEXT 262 8 "Question" }{TEXT -1 11 ": F ind " }{XPPEDIT 18 0 "Int(x/((x+1)*(x^2+1)^2),x);" "6#-%$IntG6$*&%\"x G\"\"\"*&,&F'F(F(F(F(*$,&*$F'\"\"#F(F(F(F.F(!\"\"F'" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solut ion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 60 "The integrand has a partial fraction expansion of the f orm: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/((x+1)*(x^ 2+1)^2) = A/(x+1)+(B*x+C)/(x^2+1)+(` D`*x+E)/((x^2+1)^2);" "6#/*&%\"xG \"\"\"*&,&F%F&F&F&F&*$,&*$F%\"\"#F&F&F&F,F&!\"\",(*&%\"AGF&,&F%F&F&F&F -F&*&,&*&%\"BGF&F%F&F&%\"CGF&F&,&*$F%F,F&F&F&F-F&*&,&*&%#~DGF&F%F&F&% \"EGF&F&*$,&*$F%F,F&F&F&F,F-F&" }{TEXT -1 14 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Multiplying bot h sides of equation (i) by " }{XPPEDIT 18 0 "(x+1)*(x^2+1)^2" "6#*&,&% \"xG\"\"\"F&F&F&*$,&*$F%\"\"#F&F&F&F*F&" }{TEXT -1 8 " gives: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = A*(x^2+1)^2+(B*x+ C)*(x+1)*(x^2+1)+(` D`*x+E)*(x+1);" "6#/%\"xG,(*&%\"AG\"\"\"*$,&*$F$\" \"#F(F(F(F,F(F(*(,&*&%\"BGF(F$F(F(%\"CGF(F(,&F$F(F(F(F(,&*$F$F,F(F(F(F (F(*&,&*&%#~DGF(F$F(F(%\"EGF(F(,&F$F(F(F(F(F(" }{TEXT -1 1 " " }} {PARA 257 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=A*(x^4+2*x^2 +1) + (B*x+C)*(x^3+x^2+x+1)+(` D`*x+E)*(x+1)" "6#/%\"xG,(*&%\"AG\"\"\" ,(*$F$\"\"%F(*&\"\"#F(*$F$F-F(F(F(F(F(F(*&,&*&%\"BGF(F$F(F(%\"CGF(F(,* *$F$\"\"$F(*$F$F-F(F$F(F(F(F(F(*&,&*&%#~DGF(F$F(F(%\"EGF(F(,&F$F(F(F(F (F(" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " } {TEXT 279 1 "A" }{TEXT -1 2 ", " }{TEXT 280 1 "B" }{TEXT -1 2 ", " } {TEXT 281 1 "C" }{TEXT -1 2 ", " }{TEXT 282 1 "D" }{TEXT -1 5 " and " }{TEXT 283 1 "E" }{TEXT -1 49 " by the two strategies of substituting \+ values of " }{TEXT 277 1 "x" }{TEXT -1 48 " in (ii) and equating coeff icients of powers of " }{TEXT 278 1 "x" }{TEXT -1 30 " on the left and right sides. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[co efficients*of*x^4]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'% \"xG\"\"%F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A+B;" " 6#/\"\"!,&%\"AG\"\"\"%\"BGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^3]*` . . . `;" "6#*& 7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"$F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = B+C;" "6#/\"\"!,&%\"BG\"\"\"%\"CGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficie nts*of*x^2]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\" #F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*A+B+C+`D `;" "6#/\"\"!,**&\"\"#\"\"\"%\"AGF(F(%\"BGF(%\"CGF(%#D~GF(" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of* x]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xGF'F'%(~.~.~.~ GF'" }{TEXT -1 5 " " }{XPPEDIT 18 0 "1 = B+C+` D`+E;" "6#/\"\"\",* %\"BGF$%\"CGF$%#~DGF$%\"EGF$" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[x = 0]*` . . . `;" "6#*&7#/%\"xG\"\"! \"\"\"%(~.~.~.~GF(" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A+C+E;" "6#/ \"\"!,(%\"AG\"\"\"%\"CGF'%\"EGF'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The first two equations g ive " }{XPPEDIT 18 0 "B=-A" "6#/%\"BG,$%\"AG!\"\"" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "C=-B" "6#/%\"CG,$%\"BG!\"\"" }{XPPEDIT 18 0 "``=A" " 6#/%!G%\"AG" }{TEXT -1 42 " so that the last three equations become: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2*A+`D ` = 0, ``],[` D`+E = 1, ``],[2*A+E = 0, ``]);" "6#-%*PIECEWISEG6%7$/, &*&\"\"#\"\"\"%\"AGF+F+%#D~GF+\"\"!%!G7$/,&%#~DGF+%\"EGF+F+F/7$/,&*&F* F+F,F+F+F4F+F.F/" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The equations " }{XPPEDIT 18 0 "2*A+`D ` \+ = 0" "6#/,&*&\"\"#\"\"\"%\"AGF'F'%#D~GF'\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "2*A+E = 0" "6#/,&*&\"\"#\"\"\"%\"AGF'F'%\"EGF'\"\"!" } {TEXT -1 11 " imply that" }{XPPEDIT 18 0 " ` D`-E=0" "6#/,&%#~DG\"\"\" %\"EG!\"\"\"\"!" }{TEXT -1 20 " which when added to" }{XPPEDIT 18 0 " \+ ` D`+E = 1" "6#/,&%#~DG\"\"\"%\"EGF&F&" }{TEXT -1 7 " gives " } {XPPEDIT 18 0 "2*`D ` = 1;" "6#/*&\"\"#\"\"\"%#D~GF&F&" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "We deduce that" }{XPPEDIT 18 0 " ` D` =1/2, E=1/2, A=-1/4,B=1/4" "6&/%#~DG*&\"\"\"F&\"\"#!\"\"/%\"EG*&F&F&F' F(/%\"AG,$*&F&F&\"\"%F(F(/%\"BG*&F&F&F0F(" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "C=-1/4" "6#/%\"CG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "S ubstituting these values in (i) gives the expansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/((x+1)*(x^2+1)^2) = -1/(4*(x+1)) +(x/4-1/4)/(x^2+1)+(x/2+1/2)/((x^2+1)^2)" "6#/*&%\"xG\"\"\"*&,&F%F&F&F &F&*$,&*$F%\"\"#F&F&F&F,F&!\"\",(*&F&F&*&\"\"%F&,&F%F&F&F&F&F-F-*&,&*& F%F&F1F-F&*&F&F&F1F-F-F&,&*$F%F,F&F&F&F-F&*&,&*&F%F&F,F-F&*&F&F&F,F-F& F&*$,&*$F%F,F&F&F&F,F-F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x /((x+1)*(x^2+1)^2),x) = -1/4;" "6#/-%$IntG6$*&%\"xG\"\"\"*&,&F(F)F)F)F )*$,&*$F(\"\"#F)F)F)F/F)!\"\"F(,$*&F)F)\"\"%F0F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/(x+1),x) +1/4" "6#,&-%$IntG6$*&\"\"\"F(,&%\"xGF(F (F(!\"\"F*F(*&F(F(\"\"%F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-1) /(x^2+1),x) + 1/2" "6#,&-%$IntG6$*&,&%\"xG\"\"\"F*!\"\"F*,&*$F)\"\"#F* F*F*F+F)F**&F*F*F.F+F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x+1)/(x^2 +1)^2,x)" "6#-%$IntG6$*&,&%\"xG\"\"\"F)F)F)*$,&*$F(\"\"#F)F)F)F-!\"\"F (" }{TEXT -1 16 " ------- (iii)." }}{PARA 256 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "The first term on the right side of (iii) is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-1/4" "6#,$* &\"\"\"F%\"\"%!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x+1),x) \+ = -1/4;" "6#/-%$IntG6$*&\"\"\"F(,&%\"xGF(F(F(!\"\"F*,$*&F(F(\"\"%F+F+ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+1))+c[1];" "6#,&-%#lnG6#-%$ absG6#,&%\"xG\"\"\"F,F,F,&%\"cG6#F,F," }{TEXT -1 14 " ------- (iv)." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The 2nd \+ term on the right side of (iii) is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/4" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int((x-1)/(x^2+1),x)=1/8" "6#/-%$IntG6$*&,&%\"xG\"\"\"F *!\"\"F*,&*$F)\"\"#F*F*F*F+F)*&F*F*\"\")F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*x/(x^2+1),x)-1/4" "6#,&-%$IntG6$*(\"\"#\"\"\"%\"xGF),&*$F* F(F)F)F)!\"\"F*F)*&F)F)\"\"%F-F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int( 1/(x^2+1),x)" "6#-%$IntG6$*&\"\"\"F',&*$%\"xG\"\"#F'F'F'!\"\"F*" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/ 8" "6#*&\"\"\"F$\"\")!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+1) \+ -1/4" "6#,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"F+F+F+*&F+F+\"\"%!\"\"F." } {TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x) + c[2]" "6#,&-%'arctanG6#%\"x G\"\"\"&%\"cG6#\"\"#F(" }{TEXT -1 14 " ------- (v). " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We now tackle the integ ral " }{XPPEDIT 18 0 "Int((x+1)/((x^2+1)^2),x);" "6#-%$IntG6$*&,&%\"xG \"\"\"F)F)F)*$,&*$F(\"\"#F)F)F)F-!\"\"F(" }{TEXT -1 33 " in (i) (witho ut the coefficient " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 3 " )." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In t((x+1)/((x^2+1)^2),x) = Int(x/((x^2+1)^2),x)+Int(1/((x^2+1)^2),x);" " 6#/-%$IntG6$*&,&%\"xG\"\"\"F*F*F**$,&*$F)\"\"#F*F*F*F.!\"\"F),&-F%6$*& F)F**$,&*$F)F.F*F*F*F.F/F)F*-F%6$*&F*F**$,&*$F)F.F*F*F*F.F/F)F*" } {TEXT -1 16 " ------- (vi). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "The first of the two integrals on the rig ht side of (vi) can be found by making the substitution " }{XPPEDIT 18 0 "u=x^2+1" "6#/%\"uG,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x/(x^2+1)^2,x)" "6#-%$IntG6$*&%\"xG\"\"\"*$,&*$F'\" \"#F(F(F(F,!\"\"F'" }{TEXT -1 7 " --- " }{XPPEDIT 18 0 "PIECEWISE([u = x^2+1, ``],[du = 2*x*dx, ``(1/2)*`.`*du = x*dx]);" "6#-%*PIECEWISEG 6$7$/%\"uG,&*$%\"xG\"\"#\"\"\"F-F-%!G7$/%#duG*(F,F-F+F-%#dxGF-/*(-F.6# *&F-F-F,!\"\"F-%\".GF-F1F-*&F+F-F3F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^(-2),u)" "6#-%$IntG6$)%\"uG, $\"\"#!\"\"F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= -1/(2*u) + c[3]" "6#/%!G,&*&\"\"\"F'*&\"\"#F'%\"uGF '!\"\"F+&%\"cG6#\"\"$F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x/(( x^2+1)^2),x)= -1/(2*(x^2+1))+c[3]" "6#/-%$IntG6$*&%\"xG\"\"\"*$,&*$F( \"\"#F)F)F)F-!\"\"F(,&*&F)F)*&F-F),&*$F(F-F)F)F)F)F.F.&%\"cG6#\"\"$F) " }{TEXT -1 17 " ------- (vii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 " The formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+a^2)^2),x) = x/(2*a^2*(x^2+a^2))+ 1/(2*a^3);" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(*$%\"aGF-F(F-!\" \"F,,&*&F,F(*(F-F(*$F/F-F(,&*$F,F-F(*$F/F-F(F(F0F(*&F(F(*&F-F(*$F/\"\" $F(F0F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x/a) +c" "6#,&-%'arcta nG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "gives " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/((x^2+1)^2),x) = x/(2*(x^2 +1))+1/2;" "6#/-%$IntG6$*&\"\"\"F(*$,&*$%\"xG\"\"#F(F(F(F-!\"\"F,,&*&F ,F(*&F-F(,&*$F,F-F(F(F(F(F.F(*&F(F(F-F.F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x+c[4];" "6#,&*&%'arctanG\"\"\"%\"xGF&F&&%\"cG6#\"\"%F&" }{TEXT -1 18 " ------- (viii). " }}{PARA 0 "" 0 "" {TEXT -1 28 "(vii) and (viii) imply that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "In t((x+1)/((x^2+1)^2),x) = (x-1)/(4*(x^2+1)) +1/4" "6#/-%$IntG6$*&,&%\"x G\"\"\"F*F*F**$,&*$F)\"\"#F*F*F*F.!\"\"F),&*&,&F)F*F*F/F**&\"\"%F*,&*$ F)F.F*F*F*F*F/F**&F*F*F4F/F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x +c[5]" "6#,&*&%'arctanG\"\"\"%\"xGF&F&&%\"cG6#\"\"&F&" }{TEXT -1 16 " \+ ------- (ix). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Subsituting the results from (iv), (v) and (ix) in (iii) \+ gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x/((x +1)*(x^2+1)^2),x) = -1/4;" "6#/-%$IntG6$*&%\"xG\"\"\"*&,&F(F)F)F)F)*$, &*$F(\"\"#F)F)F)F/F)!\"\"F(,$*&F)F)\"\"%F0F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/(x+1),x) +1/4" "6#,&-%$IntG6$*&\"\"\"F(,&%\"xGF(F (F(!\"\"F*F(*&F(F(\"\"%F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-1) /(x^2+1),x) + 1/2" "6#,&-%$IntG6$*&,&%\"xG\"\"\"F*!\"\"F*,&*$F)\"\"#F* F*F*F+F)F**&F*F*F.F+F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x+1)/(x^2 +1)^2,x)" "6#-%$IntG6$*&,&%\"xG\"\"\"F)F)F)*$,&*$F(\"\"#F)F)F)F-!\"\"F (" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/4;" "6#/%!G,$*&\"\"\"F'\"\"% !\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+1))+1/8;" "6#,&-%#ln G6#-%$absG6#,&%\"xG\"\"\"F,F,F,*&F,F,\"\")!\"\"F," }{TEXT -1 1 " " } {XPPEDIT 18 0 "ln(x^2+1) -1/4" "6#,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"F+F+F+ *&F+F+\"\"%!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(x) + (x-1)/ (4*(x^2+1)) + 1/4" "6#,(-%'arctanG6#%\"xG\"\"\"*&,&F'F(F(!\"\"F(*&\"\" %F(,&*$F'\"\"#F(F(F(F(F+F(*&F(F(F-F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x+c;" "6#,&*&%'arctanG\"\"\"%\"xGF&F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/4;" "6#/ %!G,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+ 1))+1/8;" "6#,&-%#lnG6#-%$absG6#,&%\"xG\"\"\"F,F,F,*&F,F,\"\")!\"\"F, " }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+1)+(x-1)/(4*(x^2+1))+c;" "6#, (-%#lnG6#,&*$%\"xG\"\"#\"\"\"F+F+F+*&,&F)F+F+!\"\"F+*&\"\"%F+,&*$F)F*F +F+F+F+F.F+%\"cGF+" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{TEXT 284 27 "___________________________" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "The fact that the terms involving \"" }{XPPEDIT 18 0 "arctan*x" "6#*&%'arctanG \"\"\"%\"xGF%" }{TEXT -1 87 "\" cancel out suggests that there is anot her way to find the integral in this question. " }}{PARA 0 "" 0 "" {TEXT -1 16 "First note that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff([x/(x^2+1)],x) = (1-x^2)/((x^2+1)^2);" "6#/-%%Diff G6$7#*&%\"xG\"\"\",&*$F)\"\"#F*F*F*!\"\"F)*&,&F*F**$F)F-F.F**$,&*$F)F- F*F*F*F-F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([1/(x^2+1)],x) = -2*x/((x^2+1)^2);" "6#/-%%DiffG6$7#*&\"\"\"F),&*$%\"xG\"\"#F)F)F)!\" \"F,,$*(F-F)F,F)*$,&*$F,F-F)F)F)F-F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "We could try to find an expansion of the form " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "x/((x+1)*(x^2+1)^2) = P/(x+1)+2*Q*x/(x^2+1)+R/(x^2+1)+S *(1-x^2)/((x^2+1)^2)-2*T*x/((x^2+1)^2);" "6#/*&%\"xG\"\"\"*&,&F%F&F&F& F&*$,&*$F%\"\"#F&F&F&F,F&!\"\",,*&%\"PGF&,&F%F&F&F&F-F&**F,F&%\"QGF&F% F&,&*$F%F,F&F&F&F-F&*&%\"RGF&,&*$F%F,F&F&F&F-F&*(%\"SGF&,&F&F&*$F%F,F- F&*$,&*$F%F,F&F&F&F,F-F&**F,F&%\"TGF&F%F&*$,&*$F%F,F&F&F&F,F-F-" } {TEXT -1 15 " ------- (x). " }}{PARA 0 "" 0 "" {TEXT -1 42 "Multiplyi ng both sides of equation (x) by " }{XPPEDIT 18 0 "(x+1)*(x^2+1)^2" "6 #*&,&%\"xG\"\"\"F&F&F&*$,&*$F%\"\"#F&F&F&F*F&" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "x = P*(x^2+1)^2+(2*Q*x+R)*(x+1)*(x^ 2+1)+S*(1-x^2)*(x+1)-2*T*x*(x+1);" "6#/%\"xG,**&%\"PG\"\"\"*$,&*$F$\" \"#F(F(F(F,F(F(*(,&*(F,F(%\"QGF(F$F(F(%\"RGF(F(,&F$F(F(F(F(,&*$F$F,F(F (F(F(F(*(%\"SGF(,&F(F(*$F$F,!\"\"F(,&F$F(F(F(F(F(**F,F(%\"TGF(F$F(,&F$ F(F(F(F(F9" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x = P*(x^4+2*x^2+1) +Q*(2*x^4+2*x^3+2*x^2+2*x)+R*(x^3+x^2+x+1)+S*(-x^3-x^2+x+1)+T*(-2*x^2- 2*x);" "6#/%\"xG,,*&%\"PG\"\"\",(*$F$\"\"%F(*&\"\"#F(*$F$F-F(F(F(F(F(F (*&%\"QGF(,**&F-F(*$F$F+F(F(*&F-F(*$F$\"\"$F(F(*&F-F(*$F$F-F(F(*&F-F(F $F(F(F(F(*&%\"RGF(,**$F$F6F(*$F$F-F(F$F(F(F(F(F(*&%\"SGF(,**$F$F6!\"\" *$F$F-FCF$F(F(F(F(F(*&%\"TGF(,&*&F-F(*$F$F-F(FC*&F-F(F$F(FCF(F(" } {TEXT -1 16 " ------- (xi). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " } {TEXT 287 1 "P" }{TEXT -1 2 ", " }{TEXT 288 1 "Q" }{TEXT -1 2 ", " } {TEXT 289 1 "R" }{TEXT -1 2 ", " }{TEXT 290 1 "S" }{TEXT -1 5 " and " }{TEXT 291 1 "T" }{TEXT -1 49 " by the two strategies of substituting \+ values of " }{TEXT 285 1 "x" }{TEXT -1 48 " in (xi) and equating coeff icients of powers of " }{TEXT 286 1 "x" }{TEXT -1 30 " on the left and right sides. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[co efficients*of*x^4]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'% \"xG\"\"%F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = P+2*Q; " "6#/\"\"!,&%\"PG\"\"\"*&\"\"#F'%\"QGF'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^3]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"$F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*Q+R-S;" "6#/\"\"!,(*&\"\"#\"\" \"%\"QGF(F(%\"RGF(%\"SG!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^2]*` . . . `;" "6#*& 7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"#F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*P+2*Q+R-S-2*T;" "6#/\"\"!,,*&\"\"#\"\"\"% \"PGF(F(*&F'F(%\"QGF(F(%\"RGF(%\"SG!\"\"*&F'F(%\"TGF(F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of *x]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xGF'F'%(~.~.~. ~GF'" }{TEXT -1 5 " " }{XPPEDIT 18 0 "1 = 2*Q+R+S-2*T;" "6#/\"\"\" ,**&\"\"#F$%\"QGF$F$%\"RGF$%\"SGF$*&F'F$%\"TGF$!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[x = 0]*` . . . `; " "6#*&7#/%\"xG\"\"!\"\"\"%(~.~.~.~GF(" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = P+R+S;" "6#/\"\"!,(%\"PG\"\"\"%\"RGF'%\"SGF'" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The f irst two equations imply that " }{XPPEDIT 18 0 "0=-P+R-S" "6#/\"\"!,(% \"PG!\"\"%\"RG\"\"\"%\"SGF'" }{TEXT -1 46 " so adding this equation to the last equation " }{XPPEDIT 18 0 "0 = P+R+S" "6#/\"\"!,(%\"PG\"\"\" %\"RGF'%\"SGF'" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "R=0" "6#/%\"RG\" \"!" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "P=-S" "6#/%\"PG,$%\"SG!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 62 "Subtracting the thi rd equation from the fourth equation gives " }{XPPEDIT 18 0 "1=-2*P+2* S" "6#/\"\"\",&*&\"\"#F$%\"PGF$!\"\"*&F'F$%\"SGF$F$" }{TEXT -1 9 " so \+ that " }{XPPEDIT 18 0 "S-P=1/2" "6#/,&%\"SG\"\"\"%\"PG!\"\"*&F&F&\"\"# F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The equations " } {XPPEDIT 18 0 "P=-S" "6#/%\"PG,$%\"SG!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "S-P=1/2" "6#/,&%\"SG\"\"\"%\"PG!\"\"*&F&F&\"\"#F(" } {TEXT -1 12 " imply that " }{XPPEDIT 18 0 "S=1/4" "6#/%\"SG*&\"\"\"F& \"\"%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "P=-1/4" "6#/%\"PG,$*& \"\"\"F'\"\"%!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "T hen " }{XPPEDIT 18 0 "Q=1/8" "6#/%\"QG*&\"\"\"F&\"\")!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "2*T=2*P+2*Q+R-S" "6#/*&\"\"#\"\"\"%\"TGF&,** &F%F&%\"PGF&F&*&F%F&%\"QGF&F&%\"RGF&%\"SG!\"\"" }{XPPEDIT 18 0 "``=-1/ 2+1/4-1/4" "6#/%!G,(*&\"\"\"F'\"\"#!\"\"F)*&F'F'\"\"%F)F'*&F'F'F+F)F) " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "T=-1/4" "6#/%\"TG,$*&\"\"\"F'\"\"%!\"\"F )" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 193 "unassign('x','P','Q','R','S','T'):\neq := x /((x+1)*(x^2+1)^2)=P/(x+1)+2*Q*x/(x^2+1)+R/(x^2+1)+\n S*(1-x ^2)/((x^2+1)^2)-2*T*x/(x^2+1)^2;\nsolve(identity(eq,x),\{P,Q,R,S,T\}); \nassign(%);\neq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/*(%\"xG\" \"\",&F'F(F(F(!\"\",&*$)F'\"\"#F(F(F(F(!\"#,,*&%\"PGF(F)F*F(**F.F(%\"Q GF(F'F(F+F*F(*&%\"RGF(F+F*F(*(%\"SGF(,&F(F(F,F*F(F+F/F(**F.F(%\"TGF(F' F(F+F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/%\"QG#\"\"\"\"\")/%\"SG #F'\"\"%/%\"PG#!\"\"F,/%\"TGF//%\"RG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%\"xG\"\"\",&F%F&F&F&!\"\",&*$)F%\"\"#F&F&F&F&!\"#,* *&F&F&*&\"\"%F&F'F&F(F(*(F1F(F%F&F)F(F&*(F1F(,&F&F&F*F(F&F)F-F&*(F,F(F %F&F)F-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/(x+1)/((x^2+1)^2) = -1/(4*(x+1))+2*x/(8*(x^2+1))+(1-x^2)/(4*(x^2+1)^2)+x/(2*(x^2+1)^2);" "6#/*(%\"xG\"\"\",&F%F&F&F&!\"\"*$,&*$F%\"\"#F&F&F&F,F(,**&F&F&*&\"\"% F&,&F%F&F&F&F&F(F(*(F,F&F%F&*&\"\")F&,&*$F%F,F&F&F&F&F(F&*&,&F&F&*$F%F ,F(F&*&F0F&*$,&*$F%F,F&F&F&F,F&F(F&*&F%F&*&F,F&*$,&*$F%F,F&F&F&F,F&F(F &" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x/((x+1)*(x^2+1)^2),x) \+ = -1/4;" "6#/-%$IntG6$*&%\"xG\"\"\"*&,&F(F)F)F)F)*$,&*$F(\"\"#F)F)F)F/ F)!\"\"F(,$*&F)F)\"\"%F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x+1 ),x) +1/8" "6#,&-%$IntG6$*&\"\"\"F(,&%\"xGF(F(F(!\"\"F*F(*&F(F(\"\")F+ F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*x/(x^2+1),x)+1/4;" "6#,&-%$I ntG6$*(\"\"#\"\"\"%\"xGF),&*$F*F(F)F)F)!\"\"F*F)*&F)F)\"\"%F-F)" } {XPPEDIT 18 0 "Int((1-x^2)/((x^2+1)^2),x)-1/4;" "6#,&-%$IntG6$*&,&\"\" \"F)*$%\"xG\"\"#!\"\"F)*$,&*$F+F,F)F)F)F,F-F+F)*&F)F)\"\"%F-F-" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((-2*x)/((x^2+1)^2),x);" "6#-%$IntG6 $*&,$*&\"\"#\"\"\"%\"xGF*!\"\"F**$,&*$F+F)F*F*F*F)F,F+" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-1/4" "6#/%!G,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+1)) + 1/8" "6#,&-%#lnG6#-%$absG6#,&%\"xG\" \"\"F,F,F,*&F,F,\"\")!\"\"F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+1 ) +x/(4*(x^2+1)) -1/(4*(x^2+1)) + c" "6#,*-%#lnG6#,&*$%\"xG\"\"#\"\"\" F+F+F+*&F)F+*&\"\"%F+,&*$F)F*F+F+F+F+!\"\"F+*&F+F+*&F.F+,&*$F)F*F+F+F+ F+F1F1%\"cGF+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as befo re. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 " Some of the steps used to find the integral in this question are given by the following Maple commands. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "Int(x/((x+1)*(x^2+1)^2),x); \n``=map(convert,%,parfrac,x);\n``=map(Int,op(1,rhs(%)),x);\n``=map(`` ,rhs(%)):\n``=value(rhs(%));\n``=map(simplify,eval(subs(``=(_u->_u),rh s(%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(%\"xG\"\"\",&F' F(F(F(!\"\",&*$)F'\"\"#F(F(F(F(!\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,(*(\"\"#!\"\",&%\"xG\"\"\"F.F.F.,&*$)F-F*F.F.F.F.!\"# F.*&F.F.*&\"\"%F.F,F.F+F+*(F5F+,&F-F.F.F+F.F/F+F.F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,(-%$IntG6$,$*(\"\"#!\"\",&%\"xG\"\"\"F/F/F/,&*$) F.F+F/F/F/F/!\"#F/F.F/-F'6$,$*&F/F/*&\"\"%F/F-F/F,F,F.F/-F'6$,$*(F9F,, &F.F/F/F,F/F0F,F/F.F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(-F$6#,& *(\"\")!\"\",&*&\"\"#\"\"\"%\"xGF/F/F.F+F/,&*$)F0F.F/F/F/F/F+F/*&#F/\" \"%F/-%'arctanG6#F0F/F/F/-F$6#,$*&#F/F6F/-%#lnG6#,&F0F/F/F/F/F+F/-F$6# ,&*&#F/F*F/-F@6#F1F/F/*&#F/F6F/F7F/F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*(\"\"%!\"\",&%\"xG\"\"\"F+F(F+,&*$)F*\"\"#F+F+F+F+F(F+*&# F+F'F+-%#lnG6#,&F*F+F+F+F+F(*&#F+\"\")F+-F36#F,F+F+" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 306 8 "Question" }{TEXT -1 11 ": Find " }{XPPEDIT 18 0 "Int( 1/(x*(x^2+1)^3),x);" "6#-%$IntG6$*&\"\"\"F'*&%\"xGF'*$,&*$F)\"\"#F'F'F '\"\"$F'!\"\"F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 307 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The integrand has a \+ partial fraction expansion of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(x*(x^2+1)^3) = A/x + (B*x+C)/(x^2+1) +(` D `*x+E)/(x^2+1)^2+(F*x+G)/(x^2+1)^3" "6#/*&\"\"\"F%*&%\"xGF%*$,&*$F'\" \"#F%F%F%\"\"$F%!\"\",**&%\"AGF%F'F-F%*&,&*&%\"BGF%F'F%F%%\"CGF%F%,&*$ F'F+F%F%F%F-F%*&,&*&%#~DGF%F'F%F%%\"EGF%F%*$,&*$F'F+F%F%F%F+F-F%*&,&*& %\"FGF%F'F%F%%\"GGF%F%*$,&*$F'F+F%F%F%F,F-F%" }{TEXT -1 15 " ------- \+ (i). " }}{PARA 0 "" 0 "" {TEXT -1 33 "Multiplying both sides of (i) by " }{XPPEDIT 18 0 "x*(x^2+1)^3" "6#*&%\"xG\"\"\"*$,&*$F$\"\"#F%F%F%\" \"$F%" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1 = A*(x^2+1)^3+(B*x+C)*x*(x^2+1)^2+(` D`*x+E)*x*(x^2+1 )+(F*x+G)*x;" "6#/\"\"\",**&%\"AGF$*$,&*$%\"xG\"\"#F$F$F$\"\"$F$F$*(,& *&%\"BGF$F+F$F$%\"CGF$F$F+F$,&*$F+F,F$F$F$F,F$*(,&*&%#~DGF$F+F$F$%\"EG F$F$F+F$,&*$F+F,F$F$F$F$F$*&,&*&%\"FGF$F+F$F$%\"GGF$F$F+F$F$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "1 = A*(x^6+3*x^4+3*x^2+1)+(B*x+C)*(x^ 5+2*x^3+x)+(` D`*x+E)*(x^3+x)+(F*x+G)*x;" "6#/\"\"\",**&%\"AGF$,**$%\" xG\"\"'F$*&\"\"$F$*$F*\"\"%F$F$*&F-F$*$F*\"\"#F$F$F$F$F$F$*&,&*&%\"BGF $F*F$F$%\"CGF$F$,(*$F*\"\"&F$*&F2F$*$F*F-F$F$F*F$F$F$*&,&*&%#~DGF$F*F$ F$%\"EGF$F$,&*$F*F-F$F*F$F$F$*&,&*&%\"FGF$F*F$F$%\"GGF$F$F*F$F$" } {TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " } {TEXT 310 1 "A" }{TEXT -1 2 ", " }{TEXT 311 1 "B" }{TEXT -1 2 ", " } {TEXT 312 1 "C" }{TEXT -1 5 " and " }{TEXT 313 1 "D" }{TEXT -1 49 " by the two strategies of substituting values of " }{TEXT 308 1 "x" } {TEXT -1 48 " in (ii) and equating coefficients of powers of " }{TEXT 309 1 "x" }{TEXT -1 30 " on the left and right sides. " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^6]*` . . . `; " "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"'F'%(~.~.~.~GF'" } {TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A+B;" "6#/\"\"!,&%\"AG\"\"\"%\"BG F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^5]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofG F'%\"xG\"\"&F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = C;" "6#/\"\"!%\"CG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[coefficients*of*x^4]*` . . . `;" "6#*&7#*(%-coefficien tsG\"\"\"%#ofGF'%\"xG\"\"%F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 3*A+2*B+`D `;" "6#/\"\"!,(*&\"\"$\"\"\"%\"AGF(F(*&\"\"#F(%\" BGF(F(%#D~GF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[coefficients*of*x^3]*` . . . `;" "6#*&7#*(%-coefficien tsG\"\"\"%#ofGF'%\"xG\"\"$F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*C+E;" "6#/\"\"!,&*&\"\"#\"\"\"%\"CGF(F(%\"EGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients *of*x^2]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"#F' %(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 3*A+B+`D `+F;" "6# /\"\"!,**&\"\"$\"\"\"%\"AGF(F(%\"BGF(%#D~GF(%\"FGF(" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x]* ` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xGF'F'%(~.~.~.~GF' " }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = C+E+G;" "6#/\"\"!,(%\"CG\"\"\" %\"EGF'%\"GGF'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "[x = 0]*` . . . `;" "6#*&7#/%\"xG\"\"!\"\"\"%(~.~.~.~GF (" }{TEXT -1 3 " " }{XPPEDIT 18 0 "1 = A;" "6#/\"\"\"%\"AG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We see immediately that " }{XPPEDIT 18 0 "C=E" "6#/%\"CG%\"EG" } {XPPEDIT 18 0 "``=G" "6#/%!G%\"GG" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"! " }{TEXT -1 8 ". Since " }{XPPEDIT 18 0 "A=1" "6#/%\"AG\"\"\"" }{TEXT -1 26 ", it quickly follows that " }{XPPEDIT 18 0 "B=`D `" "6#/%\"BG%# D~G" }{XPPEDIT 18 0 "``=F" "6#/%!G%\"FG" }{XPPEDIT 18 0 "``=-1" "6#/%! G,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence \+ (i) becomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(x *(x^2+1)^3) = 1/x-x/(x^2+1)-x/((x^2+1)^2)-x/((x^2+1)^3);" "6#/*&\"\"\" F%*&%\"xGF%*$,&*$F'\"\"#F%F%F%\"\"$F%!\"\",**&F%F%F'F-F%*&F'F%,&*$F'F+ F%F%F%F-F-*&F'F%*$,&*$F'F+F%F%F%F+F-F-*&F'F%*$,&*$F'F+F%F%F%F,F-F-" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x*(x^2+1)^3),x)=Int(1/x,x )-Int(x/(x^2+1),x)-Int(x/(x^2+1)^2,x)-Int(x/(x^2+1)^2,x)" "6#/-%$IntG6 $*&\"\"\"F(*&%\"xGF(*$,&*$F*\"\"#F(F(F(\"\"$F(!\"\"F*,*-F%6$*&F(F(F*F0 F*F(-F%6$*&F*F(,&*$F*F.F(F(F(F0F*F0-F%6$*&F*F(*$,&*$F*F.F(F(F(F.F0F*F0 -F%6$*&F*F(*$,&*$F*F.F(F(F(F.F0F*F0" }{TEXT -1 17 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "The las t three integrals on the right side of (iii) can all be found by means of the substitution " }{XPPEDIT 18 0 "u=x^2+1" "6#/%\"uG,&*$%\"xG\"\" #\"\"\"F)F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(x/(x^2+1),x)" "6#-%$IntG6$*&%\"xG\"\"\",&*$F'\"\"#F (F(F(!\"\"F'" }{TEXT -1 7 " --- " }{XPPEDIT 18 0 "PIECEWISE([u=x^2+1 ,``],[du=2*dx,``(1/2)*`.`*du=dx])" "6#-%*PIECEWISEG6$7$/%\"uG,&*$%\"xG \"\"#\"\"\"F-F-%!G7$/%#duG*&F,F-%#dxGF-/*(-F.6#*&F-F-F,!\"\"F-%\".GF-F 1F-F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/u,u)" "6#-%$IntG6$*&\"\"\"F' %\"uG!\"\"F(" }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(u)) + c[1]" "6#,&-%#lnG6#-%$absG 6#%\"uG\"\"\"&%\"cG6#F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+1) + c[1]" "6#,&-%#lnG6#,&*$%\"xG\"\"# \"\"\"F+F+F+&%\"cG6#F+F+" }{TEXT -1 16 " ------- (iv). " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x/(x^2+1)^2,x)" "6#-%$IntG6$*&%\"xG\"\"\"*$,&*$F'\"\"#F(F(F(F,! \"\"F'" }{TEXT -1 7 " --- " }{XPPEDIT 18 0 "PIECEWISE([u=x^2+1,``],[ du=2*dx,``(1/2)*`.`*du=dx])" "6#-%*PIECEWISEG6$7$/%\"uG,&*$%\"xG\"\"# \"\"\"F-F-%!G7$/%#duG*&F,F-%#dxGF-/*(-F.6#*&F-F-F,!\"\"F-%\".GF-F1F-F3 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(u^(-2),u)" "6#-%$IntG6$)%\"uG,$\"\" #!\"\"F'" }{XPPEDIT 18 0 "``=-1/(2*u) + c[2]" "6#/%!G,&*&\"\"\"F'*&\" \"#F'%\"uGF'!\"\"F+&%\"cG6#F)F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1/(2*(x^2+1)) + c[2]" "6#/%!G,&*&\" \"\"F'*&\"\"#F',&*$%\"xGF)F'F'F'F'!\"\"F-&%\"cG6#F)F'" }{TEXT -1 15 " \+ ------- (v). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x/(x^2+1)^3,x)" "6#-%$IntG6$*&%\"xG \"\"\"*$,&*$F'\"\"#F(F(F(\"\"$!\"\"F'" }{TEXT -1 7 " --- " } {XPPEDIT 18 0 "PIECEWISE([u=x^2+1,``],[du=2*dx,``(1/2)*`.`*du=dx])" "6 #-%*PIECEWISEG6$7$/%\"uG,&*$%\"xG\"\"#\"\"\"F-F-%!G7$/%#duG*&F,F-%#dxG F-/*(-F.6#*&F-F-F,!\"\"F-%\".GF-F1F-F3" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I nt(u^(-3),u)" "6#-%$IntG6$)%\"uG,$\"\"$!\"\"F'" }{XPPEDIT 18 0 "``=-1/ (4*u^2) + c[3]" "6#/%!G,&*&\"\"\"F'*&\"\"%F'*$%\"uG\"\"#F'!\"\"F-&%\"c G6#\"\"$F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=-1/(4*(x^2+1)^2) + c[3]" "6#/%!G,&*&\"\"\"F'*&\"\"%F '*$,&*$%\"xG\"\"#F'F'F'F.F'!\"\"F/&%\"cG6#\"\"$F'" }{TEXT -1 16 " --- ---- (vi). " }}{PARA 0 "" 0 "" {TEXT -1 58 "Using the results (iv), (v ) and (vi) in (iii) we see that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Int(1/(x*(x^2+1)^3),x)=ln(abs(x))-1/2" "6#/-%$IntG6$*& \"\"\"F(*&%\"xGF(*$,&*$F*\"\"#F(F(F(\"\"$F(!\"\"F*,&-%#lnG6#-%$absG6#F *F(*&F(F(F.F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+1)+1/(2*(x^2+1 ))+1/(4*(x^2+1)^2)+c;" "6#,*-%#lnG6#,&*$%\"xG\"\"#\"\"\"F+F+F+*&F+F+*& F*F+,&*$F)F*F+F+F+F+!\"\"F+*&F+F+*&\"\"%F+*$,&*$F)F*F+F+F+F*F+F0F+%\"c GF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Int(1/(x*(x^2+1)^3),x);\n``=map(convert,% ,parfrac,x);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$ IntG6$*&\"\"\"F'*&%\"xGF'),&*$)F)\"\"#F'F'F'F'\"\"$F'!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,**&\"\"\"F*%\"xG!\"\"F**&F+F*, &*$)F+\"\"#F*F*F*F*!\"$F,*&F+F*F.F,F,*&F+F*F.!\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,*-%#lnG6#%\"xG\"\"\"*&F*F**&\"\"%F*),&*$)F)\" \"#F*F*F*F*F2F*!\"\"F**&#F*F2F*-F'6#F/F*F3*&F*F**&F2F*F/F*F3F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 " " 0 "" {TEXT 298 8 "Question" }{TEXT -1 11 ": Find " }{XPPEDIT 18 0 "Int((x^4+1)/((x^2+1)^3),x);" "6#-%$IntG6$*&,&*$%\"xG\"\"%\"\"\"F+F+ F+*$,&*$F)\"\"#F+F+F+\"\"$!\"\"F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solution" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 60 "The integrand has a partial fractio n expansion of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(x^4+1)/((x^2+1)^3) = (A*x+B)/(x^2+1)+(C*x+`D `)/((x^2+ 1)^2)+(E*x+F)/((x^2+1)^3);" "6#/*&,&*$%\"xG\"\"%\"\"\"F)F)F)*$,&*$F'\" \"#F)F)F)\"\"$!\"\",(*&,&*&%\"AGF)F'F)F)%\"BGF)F),&*$F'F-F)F)F)F/F)*&, &*&%\"CGF)F'F)F)%#D~GF)F)*$,&*$F'F-F)F)F)F-F/F)*&,&*&%\"EGF)F'F)F)%\"F GF)F)*$,&*$F'F-F)F)F)F.F/F)" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 33 "Multiplying both sides of (i) by " }{XPPEDIT 18 0 "(x^2+1)^3;" "6#*$,&*$%\"xG\"\"#\"\"\"F(F(\"\"$" }{TEXT -1 8 " gives : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^4+1 = (A*x+B) *(x^2+1)^2+(C*x+` D`)*(x^2+1)+``(E*x+F);" "6#/,&*$%\"xG\"\"%\"\"\"F(F( ,(*&,&*&%\"AGF(F&F(F(%\"BGF(F(*$,&*$F&\"\"#F(F(F(F2F(F(*&,&*&%\"CGF(F& F(F(%#~DGF(F(,&*$F&F2F(F(F(F(F(-%!G6#,&*&%\"EGF(F&F(F(%\"FGF(F(" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^4+1 = (A*x+B)*(x^4+2*x^2+1)+(C *x+` D`)*(x^2+1)+``(E*x+F);" "6#/,&*$%\"xG\"\"%\"\"\"F(F(,(*&,&*&%\"AG F(F&F(F(%\"BGF(F(,(*$F&F'F(*&\"\"#F(*$F&F2F(F(F(F(F(F(*&,&*&%\"CGF(F&F (F(%#~DGF(F(,&*$F&F2F(F(F(F(F(-%!G6#,&*&%\"EGF(F&F(F(%\"FGF(F(" } {TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "We can obtain equations involving " } {TEXT 302 1 "A" }{TEXT -1 2 ", " }{TEXT 303 1 "B" }{TEXT -1 2 ", " } {TEXT 304 1 "C" }{TEXT -1 5 " and " }{TEXT 305 1 "D" }{TEXT -1 49 " by the two strategies of substituting values of " }{TEXT 300 1 "x" } {TEXT -1 48 " in (ii) and equating coefficients of powers of " }{TEXT 301 1 "x" }{TEXT -1 30 " on the left and right sides. " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^5]*` . . . `; " "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"&F'%(~.~.~.~GF'" } {TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A;" "6#/\"\"!%\"AG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of *x^4]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"%F'%(~ .~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "1 = B;" "6#/\"\"\"%\"BG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[co efficients*of*x^3]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'% \"xG\"\"$F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*A+C; " "6#/\"\"!,&*&\"\"#\"\"\"%\"AGF(F(%\"CGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x^2]*` . . . `;" "6#*&7#*(%-coefficientsG\"\"\"%#ofGF'%\"xG\"\"#F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = 2*B+`D `;" "6#/\"\"!,&*&\"\"#\" \"\"%\"BGF(F(%#D~GF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[coefficients*of*x]*` . . . `;" "6#*&7#*(%-coeffici entsG\"\"\"%#ofGF'%\"xGF'F'%(~.~.~.~GF'" }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 = A+C+E;" "6#/\"\"!,(%\"AG\"\"\"%\"CGF'%\"EGF'" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[x = 0]*` . . . ` ;" "6#*&7#/%\"xG\"\"!\"\"\"%(~.~.~.~GF(" }{TEXT -1 3 " " }{XPPEDIT 18 0 "1 = B+`D `+F;" "6#/\"\"\",(%\"BGF$%#D~GF$%\"FGF$" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We se e immediately that " }{XPPEDIT 18 0 "A = C;" "6#/%\"AG%\"CG" } {XPPEDIT 18 0 "`` = E;" "6#/%!G%\"EG" }{XPPEDIT 18 0 "``=0" "6#/%!G\" \"!" }{TEXT -1 8 ". Since " }{XPPEDIT 18 0 "B = 1;" "6#/%\"BG\"\"\"" } {TEXT -1 25 ", it quickly follows that" }{XPPEDIT 18 0 "` D` = -2;" "6 #/%#~DG,$\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F = 2;" "6#/% \"FG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "Hence (i) b ecomes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^4+1)/( (x^2+1)^3) = 1/(x^2+1)-2/((x^2+1)^2)+2/((x^2+1)^3);" "6#/*&,&*$%\"xG\" \"%\"\"\"F)F)F)*$,&*$F'\"\"#F)F)F)\"\"$!\"\",(*&F)F),&*$F'F-F)F)F)F/F) *&F-F)*$,&*$F'F-F)F)F)F-F/F/*&F-F)*$,&*$F'F-F)F)F)F.F/F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int( (x^4+1)/((x^2+1)^3),x)=Int(1/(x^2+ 1),x)-2*Int(1/((x^2+1)^2),x)+2*Int(1/(x^2+1)^3,x)" "6#/-%$IntG6$*&,&*$ %\"xG\"\"%\"\"\"F,F,F,*$,&*$F*\"\"#F,F,F,\"\"$!\"\"F*,(-F%6$*&F,F,,&*$ F*F0F,F,F,F2F*F,*&F0F,-F%6$*&F,F,*$,&*$F*F0F,F,F,F0F2F*F,F2*&F0F,-F%6$ *&F,F,*$,&*$F*F0F,F,F,F1F2F*F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ar ctan*x-2*(x/(2*(x^2+1))+arctan*x/2)+2*(x/(4*(x^2+1)^2)+3*x/(8*(x^2+1)) +3/8*arctan*x)+c" "6#/%!G,**&%'arctanG\"\"\"%\"xGF(F(*&\"\"#F(,&*&F)F( *&F+F(,&*$F)F+F(F(F(F(!\"\"F(*(F'F(F)F(F+F1F(F(F1*&F+F(,(*&F)F(*&\"\"% F(*$,&*$F)F+F(F(F(F+F(F1F(*(\"\"$F(F)F(*&\"\")F(,&*$F)F+F(F(F(F(F1F(** FF1F'F(F)F(F(F(F(%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x/ (2*(x^2+1)^2)-x/(4*(x^2+1))+3/4" "6#/%!G,(*&%\"xG\"\"\"*&\"\"#F(*$,&*$ F'F*F(F(F(F*F(!\"\"F(*&F'F(*&\"\"%F(,&*$F'F*F(F(F(F(F.F.*&\"\"$F(F1F.F (" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x + c" "6#,&*&%'arctanG\"\" \"%\"xGF&F&%\"cGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "Int((x^4+1)/(x^2+1)^3,x );\n``=map(convert,%,parfrac,x);\n``=map(Int,op(1,rhs(%)),x);\n``=map( ``,rhs(%)):\n``=value(rhs(%));\n``=map(simplify,eval(subs(``=(_u->_u), rhs(%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*$)%\"xG\" \"%\"\"\"F,F,F,F,,&*$)F*\"\"#F,F,F,F,!\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,(*&\"\"#\"\"\",&*$)%\"xGF*F+F+F+F+!\"#!\" \"*&F+F+F,F1F+*&F*F+F,!\"$F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G ,(-%$IntG6$,$*&\"\"#\"\"\",&*$)%\"xGF+F,F,F,F,!\"#!\"\"F0F,-F'6$*&F,F, F-F2F0F,-F'6$,$*&F+F,F-!\"$F,F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,(-F$6#,&*&%\"xG\"\"\",&*$)F*\"\"#F+F+F+F+!\"\"F0-%'arctanG6#F*F0F+ -F$6#F1F+-F$6#,(*(F/F0F*F+F,!\"#F+**\"\"$F+\"\"%F0F*F+F,F0F+*&#F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "T asks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 45 "In questions 1 and 2 find the given integral." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 7 " \+ " }{XPPEDIT 18 0 "Int((x-3)/((x^2+2*x+4)^2),x);" "6#-%$IntG6$*&,&%\" xG\"\"\"\"\"$!\"\"F)*$,(*$F(\"\"#F)*&F/F)F(F)F)\"\"%F)F/F+F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "- (4*x+7)/(6*(x^2+2*x+4))-2/(3*sqrt(3))" "6#,&*&,&*&\"\"%\"\"\"%\"xGF(F( \"\"(F(F(*&\"\"'F(,(*$F)\"\"#F(*&F/F(F)F(F(F'F(F(!\"\"F1*&F/F(*&\"\"$F (-%%sqrtG6#F4F(F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan((x+1)/sqrt (3)) +c" "6#,&-%'arctanG6#*&,&%\"xG\"\"\"F*F*F*-%%sqrtG6#\"\"$!\"\"F*% \"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "Int((x-3)/((x^2+2*x+4)^2),x);\n``= student[completesquare](%,x);\n``=student[changevar](x+1=u,rhs(%),u); \n``=map(expand,rhs(%));\n``=map(Int,op(1,rhs(%)),u);\n``=map(``,rhs(% )):\n``=value(rhs(%));\n``=eval(subs(``=(_u->_u),rhs(%)));\n``=map(sim plify,subs(u=x+1,rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6 $*&,&%\"xG\"\"\"\"\"$!\"\"F),(*$)F(\"\"#F)F)*&F/F)F(F)F)\"\"%F)!\"#F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&,&%\"xG\"\"\"\"\"$! \"\"F+,&*$),&F*F+F+F+\"\"#F+F+F,F+!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&,&\"\"%!\"\"%\"uG\"\"\"F-,&*$)F,\"\"#F-F-\"\"$F-! \"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,&*&\"\"%\"\"\", &*$)%\"uG\"\"#F+F+\"\"$F+!\"#!\"\"*&F,F2F/F+F+F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$IntG6$,$*&\"\"%\"\"\",&*$)%\"uG\"\"#F,F,\"\"$F ,!\"#!\"\"F0F,-F'6$*&F-F3F0F,F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G,&-F$6#,&**\"\"#\"\"\"\"\"$!\"\"%\"uGF+,&*$)F.F*F+F+F,F+F-F-*&#F*\" \"*F+*&F,#F+F*-%'arctanG6#,$*(F,F-F.F+F,F6F+F+F+F-F+-F$6#,$*&F+F+*&F*F +F/F+F-F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(**\"\"#\"\"\"\"\" $!\"\"%\"uGF(,&*$)F+F'F(F(F)F(F*F**&#F'\"\"*F(*&F)#F(F'-%'arctanG6#,$* (F)F*F+F(F)F3F(F(F(F**&F(F(*&F'F(F,F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&#\"\"#\"\"*\"\"\"*&\"\"$#F*F(-%'arctanG6#,$*(F, !\"\",&%\"xGF*F*F*F*F,F-F*F*F*F3**F(F*F,F3F4F*,(*$)F5F(F*F**&F(F*F5F*F *\"\"%F*F3F3*&F*F**&F(F*F7F*F3F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int((x-3)/((x^2+2*x+4)^2),x) ;\n``=map(simplify,value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$*&,&%\"xG\"\"\"\"\"$!\"\"F),(*$)F(\"\"#F)F)*&F/F)F(F)F)\"\"%F)!\"# F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*(\"\"'!\"\",&*&\"\"%\"\" \"%\"xGF,F,\"\"(F,F,,(*$)F-\"\"#F,F,*&F2F,F-F,F,F+F,F(F(*&#F2\"\"*F,*& \"\"$#F,F2-%'arctanG6#,$*(F8F(,&F-F,F,F,F,F8F9F,F,F,F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }} {PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "Int((x-1)^3/(x*(x ^2+1)^2),x)" "6#-%$IntG6$*&,&%\"xG\"\"\"F)!\"\"\"\"$*&F(F)*$,&*$F(\"\" #F)F)F)F0F)F*F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "(x+1)/(x^2+1)-ln(abs(x))+1/2;" "6#,(*&,&%\"xG \"\"\"F'F'F',&*$F&\"\"#F'F'F'!\"\"F'-%#lnG6#-%$absG6#F&F+*&F'F'F*F+F' " }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+1)+2*arctan*x+c" "6#,(-%#lnG6 #,&*$%\"xG\"\"#\"\"\"F+F+F+*(F*F+%'arctanGF+F)F+F+%\"cGF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "Int((x-1)^3/(x*(x^2+1)^2),x);\n``=map(convert,%,parf rac,x);\n``=map(expand,rhs(%));\n``=map(Int,op(1,rhs(%)),x);\n``=map(` `,rhs(%)):\n``=value(rhs(%));\n``=map(simplify,eval(subs(``=(_u->_u),r hs(%))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(,&%\"xG\"\"\"F )!\"\"\"\"$F(F*,&*$)F(\"\"#F)F)F)F)!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,(*&\"\"\"F*%\"xG!\"\"F,*&,&\"\"#F**&F/F*F +F*F,F*,&*$)F+F/F*F*F*F*!\"#F**&,&F+F*F*F*F*F1F,F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$,,*&\"\"\"F*%\"xG!\"\"F,*&\"\"#F*,&*$)F +F.F*F*F*F*!\"#F**(F.F*F+F*F/F2F,*&F+F*F/F,F**&F*F*F/F,F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,-%$IntG6$,$*&\"\"\"F+%\"xG!\"\"F-F,F+- F'6$,$*&\"\"#F+,&*$)F,F2F+F+F+F+!\"#F+F,F+-F'6$,$*(F2F+F,F+F3F6F-F,F+- F'6$*&F,F+F3F-F,F+-F'6$*&F+F+F3F-F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,-F$6#,$-%#lnG6#%\"xG!\"\"\"\"\"-F$6#,&*&F,F.,&*$)F,\"\"#F.F.F .F.F-F.-%'arctanGF+F.F.-F$6#*&F.F.F3F-F.-F$6#,$*&#F.F6F.-F*6#F3F.F.F.- F$6#F7F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&%\"xG\"\"\",&*$)F' \"\"#F(F(F(F(!\"\"F(*&F,F(-%'arctanG6#F'F(F(*&F(F(F)F-F(-%#lnGF1F-*&#F (F,F(-F46#F)F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }